algorithmic complexity of weakly semiregular partitioning and the representation number abstract a graph g is weakly semiregular if there are two numbers a b such that the degree of every vertex is a or b the weakly semiregular number of a graph g denoted by w r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph we present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two on the other hand we show that determining whether w r g 2 for a given bipartite graph g with at most three numbers in its degree set is np complete among other results for every tree t we show that w r t 2 log 2 t o 1 where t denotes the maximum degree of t a graph g is a d d s graph if the degree of every vertex of g lies in the interval d d s a d d 1 graph is said to be semiregular the semiregular number of a graph g denoted by s r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a semiregular graph we prove that the semiregular number of a tree t is t 2 on the other hand we show that determining whether s r g 2 for a given bipartite graph g with g 6 is np complete in the second part of the work we consider the representation number a graph g has a representation modulo r if there exists an injective map l v g z r such that vertices v and u are adjacent if and only if l u l v is relatively prime to r the representation number denoted by r e p g is the smallest r such that g has a representation modulo r narayan and urick conjectured that the determination of r e p g for an arbitrary graph g is a difficult problem 38 in this work we confirm this conjecture and show that if n p p then for any 0 there is no polynomial time 1 n 2 approximation algorithm for the computation of representation number of regular graphs with n vertices

chromatic index of graphs with no cycle with a unique chord the class c of graphs that do not contain a cycle with a unique chord was recently studied by trotignon and vuskovic in press 23 who proved for these graphs strong structure results which led to solving the recognition and vertex colouring problems in polynomial time in the present paper we investigate how these structure results can be applied to solve the edge colouring problem in the class we give computational complexity results for the edge colouring problem restricted to c and to the subclass c composed of the graphs of c that do not have a 4 hole we show that it is np complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of c with fixed degree d 3 for the subclass c we establish a dichotomy if the maximum degree is d 3 the edge colouring problem is np complete whereas if d 3 the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs a characterization that solves edge colouring problem in polynomial time we determine two subclasses of graphs in c of maximum degree 3 for which edge colouring is polynomial finally we remark that a consequence of one of our proofs is that edge colouring in np complete for r regular tripartite graphs of degree d 3 for r 3

parameterized complexity of max lifetime target coverage in wireless sensor networks max lifetime target coverage can be viewed as a family of problems where the task is to partition the sensors into groups and assign their time slots such that the coverage lifetime is maximized while satisfying some coverage requirement unfortunately these problems are np hard to gain insight into the source of the complexity we initiate a systematic parameterized complexity study of two types of max lifetime target coverage max min target coverage and max individual target coverage we first prove that both problems remain np hard even in the special cases where each target is covered by at most two sensors or each sensor can cover at most two targets by contrast restricting the number of targets reduces the complexity of the considered problems in other words they are both fixed parameter tractable fpt with respect to the parameter number of targets moreover we extend our studies to the structural parameter number k of sensors covering at least two targets positively both problems are in fpt with respect to k finally we show that max min target coverage is in fpt with respect to the combined parameters number of groups and number of targets covered by each group

dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

the parameterised complexity of list problems on graphs of bounded treewidth we consider the parameterised complexity of several list problems on graphs with parameter treewidth or pathwidth in particular we show that list edge chromatic number and list total chromatic number are fixed parameter tractable parameterised by treewidth whereas list hamilton path is w1 hard even parameterised by pathwidth these results resolve two open questions of fellows fomin lokshtanov rosamond saurabh szeider and thomassen 2011

feature adaptive motion tracking of ultrasound image sequences using a deformable mesh by exploiting the correlation of ultrasound speckle patterns that result from scattering by underlying tissue elements two dimensional tissue motion theoretically can be recovered by tracking the apparent movement of the associated speckle patterns speckle tracking however is an ill posed inverse problem because of temporal decorrelation of the speckle patterns and the inherent low signal to noise ratio of medical ultrasonic images this paper investigates the use of an adaptive deformable mesh for nonrigid tissue motion recovery from ultrasound images the nodes connecting the mesh elements are allocated adaptively to stable speckle patterns that are less susceptible to temporal decorrelation the authors use the approach of finite clement analysis in manipulating the irregular mesh elements a novel deformable block matching algorithm making use of a lagrange element for higher order description of local motion is proposed to estimate a nonrigid motion vector at each node in order to ensure that the motion estimates are admissible to a physically plausible solution the nodal displacements are regularized by minimizing the strain energy associated with the mesh deformations experiments based on ultrasound images of a tissue mimicking phantom and a muscle undergoing contraction and on computer simulations have shown that the proposed algorithm can successfully track nonrigid displacement fields

local edge colouring of yao like subgraphs of unit disk graphs the focus of the present paper is on providing a local deterministic algorithm for colouring the edges of yao like subgraphs of unit disk graphs these are geometric graphs such that for some positive integers l k the following property holds at each node v if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v we assume that the nodes are location aware i e they know their cartesian coordinates in the plane the algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant depending on k and l but not on the size of the graph number of hops measured in the original underlying unit disk graph away from it and hence the algorithm terminates in a constant number of steps the number of colours used is 2kl 1 and this is optimal for local algorithms since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm thus showing that in this class of graphs the price for locality is only one additional colour

the complexity of lsh feasibility in this paper we study the complexity of the following feasibility problem given an n n similarity matrix s as input is there a locality sensitive hash lsh for s we show that the lsh feasibility problem is np hard even in the following strong promise version either s admits an lsh or s is at l1 distance at least n2 from every similarity that admits an lsh we complement this hardness result by providing an o 3n algorithm for the lsh feasibility problem which improves upon the naive n n time algorithm we prove that this running time is tight modulo constants under the exponential time hypothesis

membership in constant time and almost minimum space this paper deals with the problem of storing a subset of elements from the bounded universe mathcal m 0 ldots m 1 so that membership queries can be performed efficiently in particular we introduce a data structure to represent a subset of n elements of mathcal m in a number of bits close to the information theoretic minimum b left lceil lg m choose n right rceil and use the structure to answer membership queries in constant time

weighted and locally bounded list colorings in split graphs cographs and partial k trees abstract for a fixed number of colors we show that in node weighted split graphs cographs and graphs of bounded tree width one can determine in polynomial time whether a proper list coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists we also show that this result is tight in some sense by providing hardness results for the cases where any one of the assumptions does not hold the edge coloring variant is also studied as well as special cases of cographs and split graphs

tree compression with top trees we introduce a new compression scheme for labeled trees based on top trees our compression scheme is the first to simultaneously take advantage of internal repeats in the tree as opposed to the classical dag compression that only exploits rooted subtree repeats while also supporting fast navigational queries directly on the compressed representation we show that the new compression scheme achieves close to optimal worst case compression can compress exponentially better than dag compression is never much worse than dag compression and supports navigational queries in logarithmic time

a simple optimal representation for balanced parentheses we consider succinct or highly space efficient representations of a static string consisting of n pairs of balanced parentheses which support natural operations such as finding the matching parenthesis for a given parenthesis or finding the pair of parentheses that most tightly enclose a given pair this problem was considered by jacobson space efficient static trees and graphs in proc of the 30th focs 1989 pp 549 554 and munro and raman succinct representation of balanced parentheses and static trees siam j comput 31 2001 762 776 who gave o n bit and 2n o n bit representations respectively that supported the above operations in o 1 time on the ram model of computation this data structure is a fundamental tool in succinct representations and has applications in representing suffix trees ordinal trees planar graphs and permutations we consider the practical performance of parenthesis representations first we give a new 2n o n bit representation that supports all the above operations in o 1 time this representation is conceptually simpler its space bound has a smaller o n term and it also has a simple and uniform o n time and space construction algorithm we implement our data structure and a variant of jacobson s and evaluate their practical performance speed and memory usage when used in a succinct representation of trees derived from xml documents as a baseline we compare our representations against a widely used implementation of the standard dom document object model representation of xml documents both succinct representations use orders of magnitude less space than dom and tree traversal operations are usually only slightly slower than in dom

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

simultaneous encodings for range and next previous larger smaller value queries given an array of n elements from a total order we propose encodings that support various range queries range minimum range maximum and their variants and previous and next smaller larger value queries when query time is not of concern we obtain a 4 088n o n bit encoding that supports all these queries for the case when we need to support all these queries in constant time we give an encoding that takes 4 585n o n bits where n is the length of input array this improves the 5 08n o n bit encoding obtained by encoding the colored 2d min and max heaps proposed by fischer 11 we first extend the original dfuds 8 encoding of the colored 2d min max heap that supports the queries in constant time then we combine the extended dfuds of 2d min heap and 2d max heap using the minmax encoding of gawrychowski and nicholson 15 with some modifications we also obtain encodings that take lesser space and support a subset of these queries

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

polyhedron of triangle free simple 2 matchings in subcubic graphs a simple 2 matching in an edge weighted graph is a subgraph all of whose nodes have degree 0 1 or 2 we consider the problem of finding a maximum weight simple 2 matching that contains no triangles which is closely related to a class of relaxations of the traveling salesman problem tsp our main results are for graphs with maximum degree 3 a complete description of the convex hull of incidence vectors of triangle free simple 2 matchings and a strongly polynomial time algorithm for the above problem our system requires the use of a type of comb inequality introduced by grotschel and padberg for the tsp polytope that has 0 1 2 coefficients and hence is more general than the well known blossom inequality used in edmonds characterization of the simple 2 matching polytope

an unstaggered colocated finite difference scheme for solving time domain maxwell s equations in curvilinear coordinates in this paper we present a new unstaggered colocated finite difference scheme for solving time domain maxwell s equations in a curvilinear coordinate system all components of the electric and magnetic fields are defined at the same spatial point a combination of one sided forward and backward difference fd bd operators for the spatial derivatives is used to produce the same order of accuracy as a staggered central differencing scheme in the temporal variable the usual leapfrog integration is used the computational domain is bounded at the far end by a curvilinear perfectly matched layer pml the pml region is terminated with a first order engquist majda type absorbing boundary condition abc a comparison is shown with results available in the literature for te sub z scattering by conducting cylinders equations are also presented for the three dimensional 3 d case

p k coloring problems in line graphs the p k coloring problems generalize the usual coloring problem by replacing stable sets by cliques and stable sets complexities of some variations of p k coloring problems split coloring and cocoloring are studied in line graphs polynomial algorithms or proofs of np completeness are given according to the complexity status we show that the most general p k coloring problems are more difficult than the cocoloring and the split coloring problems while there is no such relation between the last two problems we also give complexity results for the problem of finding a maximum p k colorable subgraph in line graphs finally upper bounds on the optimal values are derived in general graphs by sequential algorithms based on welsh powell and matula orderings

succinct representations of graphs for a fixed graph property q the complexity of the problem given a graph g does g have property q is usually investigated as a function of v the number of vertices in g with the assumption that the input size is polynomial in v in this paper the complexity of these problems is investigated when the input graph is given by a succinct representation by a succinct representation it is meant that the input size is polylog in v it is shown that graph problems which are approached this way become intractable actually no nontrivial problem could be found which can be solved in polynomial time the main result is characterizing a large class of graph properties for which the respective succinct problem is np hard trying to locate these problems within the p time hierarchy shows that the succinct versions of polynomially equivalent problems may not be polynomially equivalent

integer round up property for the chromatic number of some h perfect graphs we prove that every h perfect line graph and every t perfect claw free graph g has the integer round up property for the chromatic number for every non negative integral weight function c on the vertices of g the weighted chromatic number of g c can be obtained by rounding up its fractional relaxation as a corollary we obtain that the weighted chromatic number can be computed in polynomial time for these graphs finally we show a new example of a graph operation which preserves the integer round up property for the chromatic number and use it to provide a first example of a t perfect 3 colorable graph which does not have this property

jump number maximization for proper interval graphs and series parallel graphs in this paper the order theoretic problem of finding linear extensions of a partially ordered set poset with certain properties is mapped to the corresponding graphtheorectic problem given a poset a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the poset we want to find a topological sequence that minimizes or maximizes the total number of jumps in graph theoretic terms define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized jump number minimization problem or minimized jump number maximization problem the complexities of both problems with respect to some special graphs are investigated first in this paper we then study the jump maximization problem for proper interval graphs and series parallel graphs from this study we hope that more researches can be derived from these interdisciplinary knowledge domains

space efficient euler partition and bipartite edge coloring abstract we describe space efficient algorithms for two problems on undirected multigraphs euler partition partitioning the edges into a minimum number of trails and bipartite edge coloring coloring the edges of a bipartite multigraph with the minimum number of colors let n m and 1 be the numbers of vertices and of edges and the maximum degree respectively of the input multigraph for euler partition we reduce the amount of working memory needed by a logarithmic factor from n m log n m bits to o n m bits while preserving a running time of o n m for bipartite edge coloring still using o n m bits of working memory we achieve a running time of o n m min log log log m log this is o m log log if m n log n log log n log n to be compared with o m log for the fastest known algorithm instrumental in obtaining the latter result is a new data structure presented here the subgraph stack that allows graph algorithms centered around recursive calls on substantially smaller subgraphs to be implemented in a space efficient manner

on compact representations of all pairs shortest path distance matrices let g be an unweighted and undirected graph of n nodes and let d be the nxn matrix storing the all pairs shortest path distances in g since d contains integers in n its plain storage takes n 2log n 1 bits however a simple counting argument shows that n 2 2 bits are necessary to store d in this paper we investigate the question of finding a succinct representation of d that requires o n 2 bits of storage and still supports constant time access to each of its entries this is asymptotically optimal in the worst case and far from the information theoretic lower bound by a multiplicative factor log 23 1 585 as a result o 1 bits per pairs of nodes in g are enough to retain constant time access to their shortest path distance we achieve this result by reducing the storage of d to the succinct storage of labeled trees and ternary sequences for which we properly adapt and orchestrate the use of known compressed data structures this approach can be easily and optimally extended to graphs whose edge weights are positive integers bounded by a constant value

a fast general methodology for information theoretically optimal encodings of graphs we propose a fast methodology for encoding graphs with information theoretically minimum numbers of bits specifically a graph with property pi is called a em pi graph if pi satisfies certain properties then an n node m edge pi graph g can be encoded by a binary string x such that 1 g and x can be obtained from each other in o n log n time and 2 x has at most beta n o beta n bits for any continuous superadditive function beta n so that there are at most 2 beta n o beta n distinct n node pi graphs the methodology is applicable to general classes of graphs this paper focuses on planar graphs examples of such pi include all conjunctions over the following groups of properties 1 g is a planar graph or a plane graph 2 g is directed or undirected 3 g is triangulated triconnected biconnected merely connected or not required to be connected 4 the nodes of g are labeled with labels from 1 ldots ell 1 for ell 1 leq n 5 the edges of g are labeled with labels from 1 ldots ell 2 for ell 2 leq m and 6 each node respectively edge of g has at most ell 3 o 1 self loops respectively ell 4 o 1 multiple edges moreover ell 3 and ell 4 are not required to be o 1 for the cases of pi being a plane triangulation these examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs other than rooted trees with known polynomial time information theoretically optimal coding schemes

succinct representation of labeled trees we give a representation for labeled ordered trees that supports labeled queries such as finding the i th ancestor of a node with a given label our representation is succinct namely the redundancy is small o of the optimal space for storing the tree this improves the representation of he et al isaac 2012 8 which is succinct only when the entropy of the labels is 1

space efficient preprocessing schemes for range minimum queries on static arrays given a static array of n totally ordered objects the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on line queries of the form what is the position of a minimum element in the subarray ranging from i to j we focus on two settings where 1 the input array is available at query time and 2 the input array is available only at construction time in setting 1 we show new data structures a of size frac 2n c n theta bigl frac n lg lg n c n lg n bigr bits and query time o c n for any positive integer function c n in o bigl n varepsilon bigr for an arbitrary constant 0 varepsilon 1 or b with o nh k o n bits and o 1 query time where h k denotes the empirical entropy of k th order of the input array in setting 2 we give a data structure of size 2n o n bits and query time o 1 all data structures can be constructed in linear time and almost in place

succinct representation of balanced parentheses and static trees we consider the implementation of abstract data types for the static objects binary tree rooted ordered tree and a balanced sequence of parentheses our representations use an amount of space within a lower order term of the information theoretic minimum and support in constant time a richer set of navigational operations than has previously been considered in similar work in the case of binary trees for instance we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned the approach is applied to produce a succinct representation of planar graphs in which one can test adjacency in constant time

succinct data structure for dynamic trees with faster queries abstract navarro and sadakane 19 gave a dynamic succinct data structure for storing an ordinal tree the structure supports tree queries in either o log n log log n or o log n time and insertion or deletion of a single node in o log n time in this paper we improve the result of navarro and sadakane by reducing the time complexities of some queries e g degree and level ancestor from o log n to o log n log log n

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

decompositions for the edge colouring of reduced indifference graphs the chromatic index problem finding the minimum number of colours required for colouring the edges of a graph is still unsolved for indifference graphs whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order we present new positive evidence for the conjecture every non neighbourhood overfull indifference graph can be edge coloured with maximum degree colours two adjacent vertices are twins if they belong to the same maximal cliques a graph is reduced if it contains no pair of twin vertices a graph is overfull if the total number of edges is greater than the product of the maximum degree by n 2 where n is the number of vertices we give a structural characterization for neighbourhood overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood overfull we show that the chromatic index for all reduced indifference graphs is the maximum degree we present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours

note on nonsymmetric finite differences for maxwell s equations numerical solution methods for discretizing second derivatives are available for exam ple centered nite differences in space and nystr om or st ormer methods in time thesemethods may have advantages in accuracy and stability over their rst derivative counter parts however the imposition of boundary conditions is more dif cult in the second orderformulation

an upper bound for the choice number of star edge coloring of graphs abstract the star chromatic index of a multigraph g denoted s g is the minimum number of colors needed to properly color the edges of g such that no path or cycle of length four is bi colored a multigraph g is star k edge colorable if s g k dvo ak et al 2013 proved that every subcubic multigraph is star 7 edge colorable they conjectured in the same paper that every subcubic multigraph should be star 6 edge colorable in this paper we consider this problem in a more general setting we investigate star list edge coloring of general graph g and obtain an upper bound for the choice number of star edge coloring of graphs namely we proved that s l 2 3 2 1 2 1 2 2

algorithms for edge coloring bipartite graphs and multigraphs a minimum edge coloring of a bipartite graph is a partition of the edges into delta matchings where delta is the maximum degree in the graph coloring algorithms that run in time o min m log n 2 n 2 log n are presented the algorithms rely on an efficient procedure for the special case of delta an exact power of two the coloring algorithms can be used to find maximum cardinality matchings on regular bipartite graphs in the above time bound an algorithm for coloring multigraphs with large multiplicities is also presented

when hamming meets euclid the approximability of geometric tsp and steiner tree we prove that the traveling salesman problem sc min tsp is sf max snp hard and thus sf np hard to approximate within some constant r 1 even if all cities lie in a euclidean space of dimension log n n is the number of cities and distances are computed with respect to any lp norm the running time of recent approximation schemes for geometric sc min tsp is doubly exponential in the number of dimensions our result implies that this dependence is necessary unless np has subexponential algorithms as an intermediate step we also prove the hardness of approximating sc min tsp in hamming spaces finally we prove a similar but weaker inapproximability result for the steiner minimal tree problem sc min st the reduction for sc min tsp uses error correcting codes the reduction for sc min st uses the integrality property of sc min cut linear programming relaxations the only previous inapproximability results for metric sc min tsp involved metrics where all distances are 1 or 2

bipartite edge coloring in o delta m time we show that a minimum edge coloring of a bipartite graph can be found in o delta m time where delta and m denote the maximum degree and the number of edges of g respectively it is equivalent to finding a perfect matching in a k regular bipartite graph in o km time by sharpening the methods a minimum edge coloring of a bipartite graph can be found in o p max delta log delta m time where p max delta is the largest prime factor of delta moreover a perfect matching in a k regular bipartite graph can be found in o pmax k m time

the 3 colorability problem on graphs with maximum degree four the 3 colorability problem is known to be np complete in the class of graphs with maximum degree four on the other hand due to the celebrated theorem of brooks the problem has a polynomial time solution for graphs with maximum degree three to make the complexity gap more precise we study a family of intermediate graph classes between these two extremes and classify all of them according to the computational complexity of the problem in particular we generalize brooks s theorem in the case of 3 colorability to a larger class by showing that every connected graph in that class is 3 colorable unless it is a complete graph on four vertices

succinct representations of permutations and functions we investigate the problem of succinctly representing an arbitrary permutation p on 0 n 1 so that p k i can be computed quickly for any i and any positive or negative integer power k a representation taking 1 e nlgn o 1 bits suffices to compute arbitrary powers in constant time for any positive constant e 1 a representation taking the optimal lgn o n bits can be used to compute arbitrary powers in o lgn lglgn time we then consider the more general problem of succinctly representing an arbitrary function f n n so that f k i can be computed quickly for any i and any integer power k we give a representation that takes 1 e nlgn o 1 bits for any positive constant e 1 and computes arbitrary positive powers in constant time it can also be used to compute f k i for any negative integer k in optimal o 1 f k i time we place emphasis on the redundancy or the space beyond the information theoretic lower bound that the data structure uses in order to support operations efficiently a number of lower bounds have recently been shown on the redundancy of data structures these lower bounds confirm the space time optimality of some of our solutions furthermore the redundancy of one of our structures surpasses a recent lower bound by golynski golynski soda 2009 thus demonstrating the limitations of this lower bound

approximating the max edge coloring problem the max edge coloring problem is a natural weighted generalization of the classical edge coloring problem arising in the domain of communication systems in this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge coloring of the input graph minimizing the sum of all color classes weights we present new approximation results that improve substantially the known ones for several variants of the problem with respect to the class of the underlying graph in particular we deal with variants which are known to be np hard general and bipartite graphs or are proven to be np hard in this paper complete graphs with bi valued edge weights or whose complexity question still remains open trees

new algorithms for maximum disjoint paths based on tree likeness we study the classical mathsf np hard problems of finding maximum size subsets from given sets of k terminal pairs that can be routed via edge disjoint paths maxedp or node disjoint paths maxndp in a given graph the approximability of maxedp maxndp is currently not well understood the best known lower bound is 2 varomega sqrt log n assuming mathsf np not subseteq mathsf dtime n mathcal o log n this constitutes a significant gap to the best known approximation upper bound of mathcal o sqrt n due to chekuri et al theory comput 2 137 146 2006 and closing this gap is currently one of the big open problems in approximation algorithms in their seminal paper raghavan and thompson combinatorica 7 4 365 374 1987 introduce the technique of randomized rounding for lps their technique gives an mathcal o 1 approximation when edges or nodes may be used by mathcal o left log n log log n right paths in this paper we strengthen the fundamental results above we provide new bounds formulated in terms of the feedback vertex set number r of a graph which measures its vertex deletion distance to a forest in particular we obtain the following results for maxedp we give an mathcal o sqrt r log k r approximation algorithm up to a logarithmic factor our result strengthens the best known ratio mathcal o sqrt n due to chekuri et al as r le n further we show how to route varomega text opt pairs with congestion bounded by mathcal o log kr log log kr strengthening the bound obtained by the classic approach of raghavan and thompson for maxndp we give an algorithm that gives the optimal answer in time k r mathcal o r cdot n this is a substantial improvement on the run time of 2 kr mathcal o r cdot n which can be obtained via an algorithm by scheffler we complement these positive results by proving that maxedp is mathsf np hard even for r 1 and maxndp is mathsf w 1 hard when r is the parameter this shows that neither problem is fixed parameter tractable in r unless mathsf fpt mathsf w 1 and that our approximability results are relevant even for very small constant values of r

maximum cardinality simple 2 matchings in subcubic graphs a simple 2 matching in a graph is a subset of edges m such that every node is incident with at most two edges in m a simple 2 matching is called ck restricted k 2 if it contains no cycles of length k or less we begin by considering the problem of finding maximum cardinality simple 2 matchings in subcubic graphs i e graphs whose nodes have degree at most 3 for this problem we present a min max theorem a simple polynomial time algorithm and some other results then by generalizing an approach due to vornberger we show the following for subcubic graphs the maximum cardinality c3 and c4 restricted simple 2 matching problems are polynomial time solvable there exist min max theorems for these problems and for k 5 the maximum cardinality ck restricted simple 2 matching problems are np hard

the complexity of controlled selection we identify a class of problems called controlled selection problems and study their complexity a controlled selection problem involves a random selection of a sample from a large set subject to certain structural constraints the structural constraints model some desired relationships among the elements of the set which we want to preserve under random selection for example the set may be divided into equivalence classes and we may want to ensure that not too many elements are chosen from the same class or the set may correspond to vertices of a graph and we may want to ensure that the sample does not induce a cycle controlled selection problems arise in many disciplines including statistics mathematical programming combinatorial algorithms and resource allocation this is the first attempt to unify them we develop techniques for determining the complexity of controlled selection problems given the structural constraints we use these techniques to determine the complexity of many types of controlled selection problems we show that a controlled selection problem is polynomial time reducible to the problem of finding a maximum weight sample satisfying the structural constraints we also show that a general reduction in the opposite direction would imply that one way functions do not exist in contrast we show that most natural controlled selection problems have the same complexity within polynomial time as the problem of finding a maximum weight sample satisfying the structural constraints we also present efficient algorithms for several controlled selection problems

path queries on functions abstract let f 1 n 1 n be a function and l 1 n 1 indicate a label assigned to each element of the domain we design several compact data structures that answer various kinds of summary queries on the labels of paths in f for example we can find either the minimum label in f k i for a given i and any k 0 in a given range k 1 k 2 or the minimum label in f k i for a given i and k 0 using n lg n n lg o n lg n bits and time o n the inverse ackermann function within similar space we can count in time o lg n lg lg n the number of labels within a range and report each element with such labels in o lg n lg lg n additional time several other tradeoffs and possible queries are considered such as selection top r queries and majorities finally we consider queries that allow us navigate on the graph of the function such as the nearest common successor of two elements or the nearest successor or predecessor of an element within a range of labels

the tournament scheduling problem with absences abstract we study time scheduling problems with allowed absences as a new kind of graph coloring problem one may think of a sport tournament where each player each team is permitted a certain number t of absences we then examine how many rounds are needed to schedule the whole tournament in the worst case this upper limit depends on t and on the structure of the graph g whose edges represent the games that have to be played but also on whether or not the absences are announced before the tournament starts therefore we actually have two upper limits for the number of required rounds we have t g for pre scheduling if all absences are pre fixed and we have ol t g for on line scheduling if we have to stay flexible and deal with absences when they occur we conjecture that t g g 2 t and that ol t g g 2 t the first conjecture is stronger than the total coloring conjecture while the second is weaker than the on line list edge coloring conjecture our conjectures hold for all bipartite graphs for complete graphs we prove them partially lower and upper bounds to t g and ol t g for general multigraphs g are established too

local algorithms for edge colorings in udgs in this paper we consider two problems the edge coloring and the strong edge coloring problems on unit disk graphs udgs both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks it is well known that both problems are np complete and approximation algorithms for them have been extensively studied under the centralized model of computation centralized algorithms however are not suitable for ad hoc wireless sensor networks whose devices typically have limited resources and lack the centralized coordination we develop local distributed approximation algorithms for the edge coloring and the strong edge coloring problems on unit disk graphs for the edge coloring problem our local distributed algorithm has approximation ratio 2 and locality 50 for the strong edge coloring problem on udgs we present two local distributed algorithms with different tradeoffs between their approximation ratio and locality the first algorithm has ratio 128 and locality 22 whereas the second algorithm has ratio 10 and locality 180

bounding the maximum size of a packet radio network the author addresses the problem arising in the topology design of packet radio networks prns which use time division multiplexing and have a diameter constraint of what is the maximum number n sub c f k of users which can be contained in a diameter k prn with f time slots per frame the author assumed that users cannot transmit and receive simultaneously and cannot transmit receive more than one packet at a time this assumption implies that no two channels accessed by the same user may be assigned the same time slot it is shown that the problem of determining n sub c f k is identical to the problem of determining the largest number of vertices which can be contained in an f edge colorable directed graph with diameter k lower bounds on n sub c f k for f 2 k 3 4 5 are obtained by generating large graphs of this type the graphs are constructed and colored by simple and fast procedures which are similar for different values of f and k an extensive bibliography on the edge coloring problem is included

link scheduling in polynomial time two polynomial time algorithms are given for scheduling conversations in a spread spectrum radio network the constraint on conversations is that each station can converse with only one other station at a time the first algorithm is strongly polynomial and finds a schedule of minimum length that allows each pair of neighboring stations to converse directly for a prescribed length of time the second algorithm is designed for the situation in which messages must be relayed multiple hops the algorithm produces in polynomial time a routing vector and compatible link schedule that jointly meet a prespecified end to end demand so that the schedule has the smallest possible length

the complexity of path coloring and call scheduling abstract modern high performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests in all optical networks with wavelength division multiplexing connection requests must be assigned paths and colors wavelengths such that intersecting paths receive different colors and the goal is to minimize the number of colors used this path coloring problem is proved np hard for undirected and bidirected ring networks path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs which implies a polynomial time optimal algorithm for trees of constant degree as well as np hardness and an approximation algorithm with absolute approximation ratio 4 3 and asymptotic approximation ratio 1 1 for trees of arbitrary degree for bidirected trees path coloring is shown to be np hard even in the binary case a polynomial time optimal algorithm is given for path coloring in undirected or bidirected trees with n nodes under the assumption that the number of paths touching every single node of the tree is o log n 1 e call scheduling is the problem of assigning paths and starting times to calls in a network with bandwidth reservation such that the maximum completion time is minimized in the case of unit bandwidth requirements unit edge capacities and unit call durations call scheduling is equivalent to path coloring if either the bandwidth requirements or the call durations can be arbitrary call scheduling is shown np hard for virtually every network topology

measurement of viscoelastic properties of homogeneous soft solid using transient elastography an inverse problem approach two main questions are at the center of this paper the first one concerns the choice of a rheological model in the frequency range of transient elastography sonoelasticity or nmr elastography for soft solids 20 1000 hz transient elastography experiments based on plane shear waves that propagate in an agar gelatin phantom or in bovine muscles enable one to quantify their viscoelastic properties the comparison of these experimental results to the prediction of the two simplest rheological models indicate clearly that voigt s model is the better the second question studied in the paper deals with the feasibility of quantitative viscosity mapping using inverse problem algorithm in the ideal situation where plane shear waves propagate in a sample a simple inverse problem based on the helmholtz equation correctly retrieves both elasticity and viscosity in a more realistic situation with nonplane shear waves this simple approach fails nevertheless it is shown that quantitative viscosity mapping is s

random access to grammar compressed strings and trees grammar based compression where one replaces a long string by a small context free grammar that generates the string is a simple and powerful paradigm that captures sometimes with slight reduction in efficiency many of the popular compression schemes including the lempel ziv family run length encoding byte pair encoding sequitur and re pair in this paper we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string let s be a string of length n compressed into a context free grammar mathcal s of size n we present two representations of mathcal s achieving o log n random access time and either o n cdot alpha k n construction time and space on the pointer machine model or o n construction time and space on the ram here alpha k n is the inverse of the k th row of ackermann s function our representations also efficiently support decompression of any substring in s we can decompres

tissue elasticity reconstruction using linear perturbation method a new method to reconstruct the elastic modulus of soft tissue subjected to an external static compression is presented in this approach the newton raphson method is used to vary a finite element fe model of the elasticity equations to fit in a least squared sense a set of axial tissue displacement fields estimated using a correlation technique applied to ultrasound signals the ill conditioning of the hessian matrix is eliminated using the tikhonov regularization technique this regularization provides a compromise between fidelity to the observed data and a priori information of the solution using an echographic image formation model it is shown that the method converges within a few iterations 8 10 and that strain images artifacts which are common in elastography are significantly reduced after the resolution of the inverse problem

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

orderly spanning trees with applications we introduce and study orderly spanning trees of plane graphs this algorithmic tool generalizes canonical orderings which exist only for triconnected plane graphs although not every plane graph admits an orderly spanning tree we provide an algorithm to compute an orderly pair for any connected planar graph g consisting of an embedded planar graph h isomorphic to g and an orderly spanning tree of h we also present several applications of orderly spanning trees 1 a new constructive proof for schnyder s realizer theorem 2 the first algorithm for computing an area optimal 2 visibility drawing of a planar graph and 3 the most compact known encoding of a planar graph with o 1 time query support all algorithms in this paper run in linear time

theoretical limitations of the elastic wave equation inversion for tissue elastography this article examines the theoretical limitations of the local inversion techniques for the measurement of the tissue elasticity most of these techniques are based on the estimation of the phase speed or the algebraic inversion of a one dimensional wave equation to analyze these techniques the wave equation in an elastic continuum is revisited it is proven that in an infinite medium harmonic shear waves can travel at any phase speed greater than the classically known shear wave speed by demonstrating this for a special case with cylindrical symmetry hence in addition to the mechanical properties of the tissue the phase speed depends on the geometry of the wave as well the elastic waves in an infinite cylindrical rod are studied it is proven that multiple phase speeds can coexist for a harmonic wave at a single frequency this shows that the phase speed depends not only on the mechanical properties of the tissue but also on its shape the final conclusion is that the only way to avoid theore

a generalized class teacher model for some timetabling problems abstract we consider a theoretical model which extends the basic class teacher model of timetabling and which corresponds to some situations which occur frequently in the basic training programs of universities and schools we are given m teachers t 1 t m and n classes c 1 c n the set of classes is partitioned into p disjoint subsets g 1 g p in such a way that in addition to the lectures given by one teacher to one class there are some lectures given by one teacher to the students of all classes in group g l 1 l p such lectures will be called group lectures the number a lj of one hour group lectures which teacher t j must deliver to group g l and the number b ij of one hour class teaching which t j must give to class c i are given is there a timetable of t hours or length t so that each class c i and each group g l receive all their lectures but no student is scheduled to be taught by more than one teacher in each hour and no teacher must teach to more than one group or class in each hour we show that this problem is np complete and find some sufficient conditions for the existence of a timetable of length t we also describe an algorithm for constructing a timetable corresponding to the requirement matrices a a lj and b b ij and show that under a natural assumption on a and b this algorithm finds a timetable within 7 6 of the optimum length

fourier analysis of numerical algorithms for the maxwell equations the fourier method is used to analyze the dispersive dissipative and isotropy errors of various spatial and time discretizations applied to the maxwell equations on multi dimensional grids both cartesian grids and non cartesian grids based on hexagons and tetradecahedra are studied and compared the numerical errors are quantitatively determined in terms of phase speed wavenumber propagation direction gridspacings and cfl number the study shows that centered schemes are more efficient and accurate than upwind schemes and the non cartesian grids yield superior isotropy than the cartesian ones for the centered schemes the staggered grids produce less errors than the unstaggered ones a new unstaggered algorithm which has all the best properties is introduced using an optimization technique to determine the nodal weights the new algorithm provides the highest accuracy among all the schemes discussed the study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization

combined data structure for previous and next smaller values let a be a static array storing n elements from a totally ordered set we present a data structure of optimal size at most nlog 2 3 22 o n bits that allows us to answer the following queries on a in constant time without accessing a 1 previous smaller value queries where given an index i we wish to find the first index to the left of i where a is strictly smaller than at i and 2 next smaller value queries which search to the right of i as an additional bonus our data structure also allows one to answer a third kind of query given indices i

on excessive index of certain networks matching is one of the most extensively studied areas in computer science and is interesting from the combinatorial point of view as well a matching in a graph g v e is a subset m of edges no two of which have a vertex in common a matching m is said to be perfect if every vertex in g is an endpoint of one of the edges in m the excessive index of a graph g is the minimum number of perfect matchings to cover the edge set of g the study of excessive index has a number of applications particularly in scheduling theory in this paper we determine the excessive index for certain classes of graphs including augmented butterfly network and honeycomb network we also prove that the excessive index does not exist for butterfly and benes networks

minimizing makespan in no wait job shops in this paper we study polynomial time approximation schemes ptases for the no wait job shop scheduling problem with the makespan objective function it is known that the problem is maxsnp hard in the case when each job is allowed to have three operations or more we show that if each job has at most two operations the problem admits a ptas if the number of machines is a constant i e not part of the input if the number of machines is not a constant we show that the problem is hard to approximate within a factor better than 5 4

algorithmic complexity of proper labeling problems a proper labeling of a graph is an assignment of integers to some elements of a graph which may be the vertices the edges or both of them such that we obtain a proper vertex coloring via the labeling subject to some conditions the problem of proper labeling offers many variants and received a great interest during recent years we consider the algorithmic complexity of some variants of the proper labeling problems we present some polynomial time algorithms and np completeness results for them

chromatic gallai identities operating on lovasz number if g is a triangle free graph then two gallai identities can be written as alpha g overline chi l g v g alpha l g overline chi g where alpha and overline chi denote the stability number and the clique partition number and l g is the line graph of g we show that surprisingly both equalities can be preserved for any graph g by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs according to any acyclic orientation of g as a consequence one obtains an operator phi which associates to any graph parameter beta such that alpha g le beta g le overline chi g for all graph g a graph parameter phi beta such that alpha g le phi beta g le overline chi g for all graph g we prove that vartheta g le phi vartheta g and that phi overline chi f g le overline chi f g for all graph g where vartheta is lovasz theta function and overline chi f is the fractional clique partition number moreover overline chi f g le phi vartheta g for triangle free g comparing to the previous strengthenings psi vartheta and vartheta triangle of vartheta numerical experiments show that phi vartheta is a significant better lower bound for overline chi than vartheta

densities matchings and fractional edge colorings given a multigraph g v e with a positive rational weight w e on each edge e the weighted density problem wdp is to find a subset u of v with u ge 3 and odd that maximizes 2w

succinct data structures for searchable partial sums with optimal worst case performance abstract the notion of succinct indexes can be dated back from the debut of jacobson s thesis 1988 14 and has triggered many results in the last decade in traditional indexing some given data are preprocessed so as to support online queries and updates on the data as efficiently as possible when succinctness is involved we are restricted to index the data using only an information theoretically minimum number of bits this paper concerns the succinct indexing schemes for a well studied problem called searchable partial sums sps in sps an array a of n non negative k bit integers is preprocessed so as to support online sum and search queries and possibly update operation of individual entry a succinct indexing scheme would allow only k n o k n bits to represent the array a the only known result is that when k 1 in this case it is known as the dynamic bit array problem we can support both queries in o log b n time and update in o b amortized time for any b with lg n lg lg n b n this paper shows that even for k o lg lg n we can index a succinctly such that both query and update operations can be supported using the same time complexities moreover the time for update becomes the worst case time furthermore the tradeoff between the query times and the update time is optimal as implied by pa trascu and demaine s lower bound result 2006 24 in general when k o lg u we show a lower bound of lg n lg lg n time for the search query irrespective of the update time this gives a tighter lower bound as compared to that of pa trascu and demaine s which is a consequence of the requirement of succinctness on the other hand we give a succinct index that can support sum in o log b n time search in o log b n time and update in o b time where min lg lg n lg lg u lg lg lg u lg n lg lg n the query times are optimal when b n this paper also extends the searchable partial sums with insert and delete operations and provides a succinct data structure for some cases

edge colouring of regular graphs of large degree we consider the following conjecture letgbe ak regular simple graph with an even numbernof vertices ifk n 2thengisk edge colourable we show that this conjecture is true for graphs that are join of two graphs and we provide a polynomial time algorithm for finding a k edge colouring of these graphs

succinct representations of planar maps this paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible as opposed to raw compression issues the focus here is on designing data structures that preserve the possibility of answering incidence queries in constant time we propose in particular the first optimal representations for 3 connected planar graphs and triangulations which are the most standard classes of graphs underlying meshes with spherical topology optimal means that these representations asymptotically match the respective entropy of the two classes namely 2 bits per edge for 3 connected planar graphs and 1 62 bits per triangle or equivalently 3 24 bits per vertex for triangulations these representations support adjacency queries between vertices and faces in constant time

computational complexity of covering three vertex multigraphs a covering projection from a graph g onto a graph h is a mapping of the vertices of g onto the vertices of h such that for every vertex v of g the neighborhood of v is mapped bijectively onto the neighborhood of its image moreover if g and h are multigraphs then this local bijection has to preserve multiplicities of the neighbors as well the notion of covering projection stems from topology but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs it provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved we investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph g to a fixed target graph h among other partial results this problem has been shown np hard for simple regular graphs h of valency greater than 2 and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices we extend the previously known results to the ternary case i e we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices we show that even in this case a p np completeness dichotomy holds

algorithmic problems in right angled artin groups complexity and applications abstract in this paper we consider several classical and novel algorithmic problems for right angled artin groups some of which are closely related to graph theoretic problems and study their computational complexity we study these problems with a view towards applications to cryptography

efficient and compact representations of prefix codes most of the attention in statistical compression is given to the space used by the compressed sequence a problem completely solved with optimal prefix codes however in many applications the storage space used to represent the prefix code itself can be an issue in this paper we introduce and compare several techniques to store prefix codes let n be the sequence length and n be the alphabet size then a naive storage of an optimal prefix code uses mathcal o hspace 5ex left n log n right bits our first technique shows how to use mathcal o hspace 5ex left n log log n n right bits to store the optimal prefix code then we introduce an approximate technique that for any 0 takes mathcal o hspace 5ex left n log log 1 epsilon right bits to store a prefix code with an average codeword length within an additive epsilon of the minimum finally a second approximation takes for any constant c 1 vphantom sum r r mathcal o n 1 c log n bits to store a prefix code with an average codeword length at most c times the minimum in all cases our data structures allow encoding and decoding of any symbol in mathcal o hspace 5ex left 1 right time we experimentally compare our new techniques with the state of the art showing that we achieve sixfold to eightfold space reductions at the price of a slower encoding 2 5 8 times slower and decoding 12 24 times slower the approximations further reduce this space and improve the time significantly up to recovering the speed of classical implementations for a moderate penalty in the average code length as a byproduct we compare various heuristic approximate and optimal algorithms to generate length restricted codes showing that the optimal ones are clearly superior and practical enough to be implemented

an approximation algorithm for diagnostic test scheduling in multicomputer systems the problem of diagnostic test scheduling dts is to assign to each edge e of a diagnostic graph g a time interval of length l e so that intervals corresponding to edges at any given vertex do not overlap and the overall finishing time is minimum in this correspondence we show that the dts problem is np complete then we present a longest first sequential scheduling algorithm which runs in worst case time o dm log n and uses o m space to produce a solution of length less than four times optimal then we show that the general performance bound can be strengthened to 3 opt g for low degree graphs and to 2 opt g in some special cases of binomial diagnostic graphs

computing clique and chromatic number of circular perfect graphs in polynomial time a main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time grotschel et al in combinatorica 1 2 169 197 1981 perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one we consider a well studied superclass of perfect graphs satisfying this property the circular perfect graphs and show that for such graphs both clique and chromatic number are computable in polynomial time as well in addition we discuss the polynomial time computability of further graph parameters for certain subclasses of circular perfect graphs all the results strongly rely upon lovasz s theta function

analysis of the error for approximations to systems of hyperbolic equations analysis of the error for approximations to systems of hyperbolic equations d gottlieb e turkel and s abarbanel division of applied mathematics brown university providence rhode island 02912 and school of mathematical sciences sackler faculty of exact sciences tel aviv university ramat aviv tel aviv 69978 israel e mail dig cfm brown edu turkel math tau ac il saul math tau ac il

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

h coloring degree bounded acyclic digraphs an np complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number but remain np complete if the bound is higher we investigate an analogous phenomenon for digraphs focusing on the three smallest digraphs h with np complete h colorability problems it turns out that in all three cases the h coloring problem is polynomial time solvable for digraphs with in degrees at most 1 regardless of the out degree bound respectively with out degrees at most 1 regardless of the in degree bound on the other hand as soon as both in and out degrees are bounded by constants greater than or equal to 2 all three problems are again np complete interestingly the situation changes when we further restrict the inputs to be acyclic digraphs in two of the three cases the np complete problems remain np complete in the third case the h coloring problem turns out to be polynomial time solvable for acyclic digraphs with in degrees at most 2 regardless of the out degree bound respectively with out degrees at most 2 regardless of the in degree bound we also show that in this case as soon as both in and out degrees are bounded by constants greater than or equal to 3 the h coloring problem is once again np complete even for acyclic digraphs a conjecture proposed for graphs h by feder hell and huang states that any variant of the h coloring problem which is np complete without degree constraints is also np complete with degree constraints provided the degree bounds are high enough it has been verified for h coloring of digraphs but the degree bounds are quite high and are stated in terms of the sum of in and out degrees our results confirm the conjecture with the best possible bounds that are needed for np completeness moreover they underscore the fact that the bound must apply separately to both the in degrees and the out degrees

the np completeness of chromatic index in triangle free graphs with maximum vertex of degree 3 we restrict the known case about the complexity of edge coloring for cubic graphs in the case of triangle free graphs with maximum vertex degree 3

on edge colouring indifference graphs copyright c 1997 elsevier science b v all rights reserved vizingrs theorem states that the chromatic index g of a graph g is either the maximum degree d g or d g 1 a graph g is called overfull if ve g v d g vv g v 2 a sufficient condition for g d g 1 is that g contains an overfull subgraph h with d h d g plantholt proved that this condition is necessary for graphs with a universal vertex in this paper we conjecture that for indifference graphs this is also true as supporting evidence we prove this conjecture for general graphs with three maximal cliques and with no universal vertex and for indifference graphs with odd maximum degree for the latter subclass we prove that d