pinecone/core-2020-05-10-deduplication huggingface dataset:
Strong Convergence and a Game of Numbers

AbstractS. Mozes investigated a certain solitary game played on a weighted graph. Numbers are placed on the nodes of the graph, and a move consists of changing the sign of a negative number and changing the numbers on the neighboring nodes according to the weights on the edges. Mozes proved that the game has a strong convergence property when the edges have certain positive integer weights. However, his approach would give no information in the case of other weights. In this paper we first prove that strong convergence is equivalent to the fact that the game has as a certain ‘polygon property’. We can then, in a rather elementary way, characterize the assignments of weights that imply the polygon property, and hence strong convergence. Finally, we make a natural generalization of the game, where we also have weights on the nodes. The conditions for strong convergence generalize nicely to this game
A Property on Edge-disjoint Spanning Trees

AbstractLetGbe a simple graph withnvertices and letGcdenote the complement ofG. Letω(G)denote the number of components ofGandG(E) the spanning subgraph ofGwith edge setE. Suppose that |E|=k(|V(G) |-1). Consider the partition Π=〈X1,X2,...,Xk〉 ofE(G) such that |Xi|=n-1 (1≤i≤k). Defineϵ( Π)=Σki=1ω(G(Xi))-kandϵ(G)=minϵ(Π),where the minimum is taken over all such partitions. In [Europ. J. Combin.7(1986), 263-270], Payan conjectures that ifϵ(G)>0, then there exist edgese∈E(G) ande′∈E(Gc)such thatϵ(G-e+e′)<ϵ(G). This conjecture will be proved in this note
Spectra of Regular Graphs and Hypergraphs and Orthogonal Polynomials

AbstractIn this paper we study the distribution of eigenvalues of regular graphs, regular hypergraphs, and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’. Our estimates are tight for Biggs graphs and generalized polygons. We also give an application to the distribution of eigenvalues of Hecke operators acting on weight 2 cusp forms for certain congruence subgroups
Dominating Sets in Planar Graphs

AbstractMotivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Specifically, we wish to find the smallestεsuch that, fornsufficiently large, everyn-vertex planar graph contains a dominating set of size at mostεn.We prove that 1/4<ε<1/3, and we conjecture thatε=1/4.For triangulated discs we obtain a tight bound ofε=1/3.The upper bound proof yields a linear-time algorithm for finding an(n/3)-size dominating set
Moves on Pseudoline Diagrams

AbstractPseudoline diagrams are simple arrangements of pseudolines in the affine plane where at each vertex one indicates which pseudolinecrosses overthe other. They naturally appear as projections of configurations of lines in 3-space. Motivated by isotopies of these spatial configurations, we define an equivalence for diagrams, generated by two types of moves. We encode diagrams by words and consider the associated word problem. Furthermore, in this combinational framework we obtain two new results on thechiralityof triples of lines, which are useful in the study of isotopies of line configurations in 3-space
A New Bijection Between Ordered Trees and Legal Bracketings

AbstractThe Catalan numbers[formula]are known to enumerate the legal bracketings of lengthn [1]as well as the ordered trees withn+1vertices. There exists a classical bijection (cf. [2]) between these objects that will be denoted byGthroughout the paper.We give a new bijectionFbetween the same objects that has some interesting combinatorial properties. In particular,Ftransforms an operation on the legal bracketings introduced by Kreweras [3] into a simple operation on trees.In addition, we prove that the productG·Fr-1is the bijectionWdiscovered by Vaille (cited in [5]).FandGhelp to derive some combinatorial properties ofWin a straightforward manner
Extremal Permutations with Respect to Weak Majorizations

AbstractLetZ=(z1,z2,.. .,zn)denote a permutation of ann-set. DefineFl(Z)={f(z1,z2),..., f(zn-1,zn)}and its cyclic versionFc(Z)={f(z1, z2),...,f (zn-1,zn), f(zn,z1)},wheref(x,y)increases in max{x,y}and decreases in min{x,y}.We give conditions onfsuch that extremal permutations with respect to weak majorization can be found. We then use the weak majorization property to obtain extremal permutations forLf(Z)=g(f(z1,z2), ...,f(zn-1, zn))andCf(Z)=g(f(z1, z2),..., f(zn-1,zn), f(zn,z1))whengis either convex increasing or concave decreasing
Flag-transitive Extensions of Dual Affine Planes

AbstractIn this paper I finish the work carried on in [10] and [1], obtaining a classification of flag-transitive geometries belonging to the following diagram:(with1<q<∞).An infinite family of simply connected examples for this diagram withq=2is obtained by truncating Coxeter complexes of typeDn.Two different finite families with three members andq=4arise from the Steiner systems for the Mathieu groupsM22, M23andM24.The geometries of these two families are simply connected. Three more simply connected examples of rank 3 are described in [1], withq=2, 4and 16 respectively.We shall prove that no more simply connected examples exist besides those mentioned above
A Generalization of a Theorem of Diderrich in Additive Group Theory to Vertex-transitive Graphs

AbstractLetX1,X2,... ,Xkbe a family of vertex-transitive graphs without triangles on the same vertex-set. We consider the set Λ of vertices which can be reached from a given vertexaby a pathe1· ··et, whereeiis an edge ofXrj(i)with1≤j(1)<j(2) <···<j(t)≤kand 0≤t≤k(fort=0, Λis equal to the singleton {a}).Then, either Λ contains one of the connected components ofain one of theXi’s, or the cardinality of Λ is at least 1 plus the sum of thekoutdegrees.This result, applied to groups, is a refinement of a theorem of Diderrich in additive group theory [2]
Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

AbstractIn a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice.Perhaps more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs
The r-cubical Lattice and a Generalization of the cd-index

AbstractIn this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtill's work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all theri's are greater than one, is the odd alternating ranks, {1, 3, 5, ...}
Planarity and Edge Poset Dimension

AbstractDifferent areas of discrete mathematics lead to instrinsically different characterizations of planar graphs. Planarity is expressed in terms of topology, combinatorics, algebra or search trees. More recently, Schnyder's work has related planarity to partial order theory. Acyclic orientations and associated edge partial orders lead to a new characterization of planar graphs, which also describes all of the possible planar embeddings. We prove here that there is a bijection between bipolar plane digraphs and 2-dimensionalN-free partial orders. We give also a characterization of planarity in terms of 2-colorability of a graph and provide a short proof of a previous result on planar lattices
Transitive Arcs in Planes of Even Order

AbstractWhen one considers the hyperovals inPG(2,q),qeven,q>2, then the hyperoval inPG(2, 4) and the Lunelli-Sce hyperoval inPG(2, 16) are the only hyperovals stabilized by a transitive projective group [10]. In both cases, this group is an irreducible group fixing no triangle in the plane of the hyperoval, nor in a cubic extension of that plane. Using Hartley's classification of subgroups ofPGL3(q),qeven [6], allk-arcs inPG(2,q) fixed by a transitive irreducible group, fixing no triangle inPG(2,q) or inPG(2,q3), are determined. This leads to new 18-, 36- and 72-arcs inPG(2,q),q=22h. The projective equivalences among the arcs are investigated and each section ends with a detailed study of the collineation groups of these arcs
Thek-Extensions of some New Mahonian Statistics

AbstractIn previous work by the authors, new Mahonian statistics ENV, MAD and MAK were defined on words, and it was shown that ENV is equal to the classical statistic INV and that the triple statistics (des, MAK, MAD) and (exc, DEN, ENV) are equidistributed over any rearrangement class of words. Here, exc and des are the classical Eulerian statistics, while DEN is Denert's statistic. In addition, a bijection between the symmetric group and sets of weighted Motzkin paths was used to give a continued fraction expression for the generating function of (exc, INV) or (des, MAD) on the symmetric group. These results are extended to the case in which the letters of the alphabet used are divided into two classes—large and small—with corresponding changes to the definitions of the above statistics
A Coloring Problem in Hamming Spaces

AbstractTheR-domatic number of a graph is the maximum number of colors that can be used to color the vertices of the graph so that all vertices of the graph have at least one vertex of each color within distanceR.In this paper the problem of determining theR-domatic number of then-cube,P(n,R), is considered. The valueP(6,1) =5 is settled, and a conjecture by Laborde thatP(n, 1)≈n asntends to infinity is proved. Best known upper and lower bounds on theR-domatic number of then-cube are given forn≤16 andR≤7
Icosahedral Sets in PG(5, 2)

AbstractStarting out from the 15 pairs of opposite edges and the 20 faces of a coloured icosahedron, a simple new construction is given of a ‘double-five’ of planes inPG(5, 2). This last is a recently discovered configuration consisting of a setψof (15+20=)35 points inPG(5, 2) which admits two distinct decompositionsψ=α1∪α2∪α3∪α4∪α5=β1∪β2∪β3∪β4∪β5into a set of five mutually skew planes. Moreover,Ir=αr∩βris a line, for eachr,whilenrs=αr∩βsis a point, forr≠s.The new construction illuminates why the symmetry group ofψis isomorphic toA5×Z2. The setψis a set of hyperbolic type, and it has a cubic equation.Use instead of the 15 pairs of opposite edges and the 12 vertices of the icosahedron yields a setφ⊂PG(5, 2) of (15+12=)27 points. The setφis of elliptic type, but, likeψ,has cubic equation and icosahedral symmetry. The setsψandφare of Tonchev type 3b; see [15, Table I]
Strongly Regular Square-free Graphs with μ=2

AbstractBy a square in an undirected graph Γ, we mean a cyclex, y, z, wsuch thatxis not adjacent tozandyis not adjacent tow.Suppose that Γ is a strongly regular graph with μ=2, and assume that Γ does not contain a square. Pick any vertexxof Γ and let Γ′ denote the induced subgraph on the first subsconstituent of Γ with respect tox.It is known that Γ′ is strongly regular. We express the parameters of Γ and Γ′ in terms of two integer variables,pandh,which do not depend onx.We present feasibility conditions which must be satisfied bypandh,and show that, ifpsatisfies these, thenp=(f2i+1)2for some integeri(0≤i<∞), wheref0,f1,f2,... is the Fibonacci sequence, defined byf0=0,f1=1 andfj=fj-1+fj-2(2≤j≤∞)
Regular Two-graphs from the Even Unimodular Lattice E8⊕E8

AbstractStarting from the even unimodular latticeE8⊕E8,one constructs odd systems (i.e. sets of vectors with odd inner products) of 546 vectors using results of Deza and Grishukhin. One studies the subsystems consisting of 36 pairs of opposite vectors spanning equiangular lines. These subsystems represent regular two-graphs. This gives 100 such two-graphs and they coincide with the first 100 in a list of 227 two-graphs generated by E. Spence. Using the root systems of the sublattices generated by the 100 odd systems, the set of the 100 two-graphs is divided into seven classes. The first four classes correspond to the 23 Steiner triple system on 15 points containing a head, i.e. a Fano plane
The Theory of Quality Translations with Applications to Tilings

AbstractIn this paper, we consider a broad class of simply connected complexes that includes, for example, the face-to-face tilings ofEnandSn.Along with a complex (or tiling) we consider a set Q ofqualitiesthat individually can be assigned to the various cells or tiles of the complex, and a group G which permutes the elements of Q. These qualities might be colours, a set of positive real numbers or a set of affine functions. If each ordered pair of adjacent cells is associated with a group elementg∈G,and a qualityq∈ Ais assigned to any particular cell of the complex, qualities can be assigned to adjacent cells using the action of the group G on Q. In fact, since the complex is assumed to be simply connected, a quality can be assigned to any cell of the complex bytranslatingqualities along paths of adjacent cells using the group action. Our main theorem gives sufficient conditions that the quality assigned to a cell in this manner is independent of the particular path used for thetranslation,and is therefore uniquely determined. As applications of this theorem we establish ann-dimensional generalization of the converse of Maxwell ’s theorem on frameworks [10,11], we obtain some theorems on colouring the cells of a complex, and we generalize the main part of Voronoi’s famous theorem on primitive parallelohedra [16]. See [7] and the survey [1]
A Combinatorial Geometric Structure on the Space of Orders of a Field: I

AbstractWe introduce a structure of combinatorial geometry on the setχ(K)of orders of an orderable fieldK.It verifies finiteness conditions allowing us to develop a smooth combinatorial theory of the spacesχ(K)regardless of whether they are finite or infinite. It is shown that the geometriesχ(K)are binary and bipartite. We introduce the combinatorial notion of afanand use it to study binary, bipartite combinatorial geometries in general. We characterize the fieldsKfor which the geometryχ(K)is unimodular; this extends a classical result of Bröcker in quadratic form theory
A Remark on Generalized Fisher Inequalities of Delsarte'st-Designs

AbstractP. Delsarte defined the notion oft-designs inQ-polynomial association schemes and gave a Fisher type bound using linear programming. For each classical association scheme there is a geometrical interpretation of Delsarte'st-designs by a well-defined indexλ≥1,the caseλ =1giving a lower bound on the cardinality of at-design (the Singleton bound). In this paper we compare these two bounds and discuss the non-existence of tightt-designs or the lower bound of the indexλfor association schemes on sesquilinear forms
(0,1)-Matrices with No Half–Half Submatrix of Ones

AbstractWe consider the minimum number of zeroes in a 2m× 2n(0,1)-matrixMthat contains nom×nsubmatrix of ones. We show that this number, denoted byf(m,n), is at least 2n+m+ 1 form≤n. We determine exactly when this bound is sharp and determine the extremal matrices in these cases. For anym, the bound is sharp forn=mand for all but finitely manyn>m. A general upper bound due to Gentry,f(m,n) ≤ 2m+ 2n−gcd(m,n) + 1, is also derived. Our problem is a special case of the well-known Zarankiewicz problem
R-Trees and Symmetric Differences of Sets

AbstractThe supremum of the symmetric differencex▵y:= (x\y)∪(y\x) of subsetsx,yof R satisfies the so-called four-point condition; that is, for allx,x′,y,y′ ⊆ R, one has sup (x▵x′) + sup(y▵y′)≤ max { sup(x▵y) + sup(x′ ▵y′), sup (x▵y′) + sup(x′ ▵y)}. It follows that the setEof all subsets of R which are bounded from above forms a valuated matroid relative to the mapv:E×E→{-∞}∪R : (x,y)→sup (x▵y). Hence, according to T-theory, there exists an R-treeT(E,v)uniquely determined by (E,v) up to isometry, the ends of which correspond in a one-to-one fashion to the elements of the completion of (E,v). In addition, the points ofT(E,v)can be identified with those bounded subsets of R which contain their infimum.Here, we show that these observations hold true in a much more general setting: given an arbitrary non-empty setBand an arbitrary mapr:B→{-∞}∪R, the map P(B) × P(B)→R∪{±∞}: (x,y)→supr(x▵y) again satisfies the four-point condition; so any non-empty setZof subsets ofBwith supr(x▵y)<∞ for allx,y∈Zforms a valuated matroid of rank 2 relative to this map and, therefore, gives rise to an R-tree.It is shown here that every valuated matroid of rank 2 can be realized in this way by choosing an appropriate system (B,r:B→{-∞}∪R ,Z⊆ P(B)); consequently, since every R-tree can be embedded isometrically intoT(E,v)for some valuated matroid (E,v), every R-tree can, in principle, be described in terms of such a system
Circle Packings of Maps in Polynomial Time

AbstractThe Andreev–Koebe–Thurston circle packing theorem is generalized and improved in two ways. First, we obtain simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal–dual circle packing representation in a surface of constant curvature is that its universal cover is 3-connected (the map has no ‘planar’ 2-separations). Second, an algorithm is obtained that given a mapMand a rational number ϵ < 0 finds an ϵ-approximation for the radii and the co-ordinates of the centres for the primal–dual circle packing representation ofM. The algorithm is polynomial in |E(M)| and log(1/ϵ). In particular, for a map without planar 2-separations on an arbitrary surface, we have a polynomial time algorithm for simultaneous geodesic convex representations of the map and its dual, so that only edges dual to each other cross, and the angles at the crossings are arbitrarily close to π/2
Regular Maps from Voltage Assignments and Exponent Groups

AbstractIn the paper is developed a common generalization of two methods of construction of regular maps on surfaces. The first one produces graph covering projections that extend to coverings of regular embeddings of the graphs involved. The second method employs a double covering projection of graphs which, in general, need not be extendable to a covering of regular maps. In a more general approach, the latter property remains preserved but the multiplicity of the graph covering may be arbitrary. As an application, some new regular embeddings ofn-cubes and complete bipartite graphs will be constructed. Several open problems are included
Strongly Signable and Partitionable Posets

AbstractThe class ofStrongly Signablepartially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs' recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flagh-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flagh-polynomial of a simplicial complex in terms of itsh-vector is given, and is used to demonstrate that the flagh-vector is symmetric or non-negative whenever theh-vector is
Regular Perfect Systems of Sets of Iterated Differences

AbstractFors≥2, a set {a(i,j):1 ≤j≤s+1 −i≤s} wherea(1,j), 1≤ j≤s, are some prescribed integers anda(i+1,j) =|a(i,j) −a(i,j+1)|, for 1≤i<sand 1≤j≤s−i, is called a set of iterated differences. Such a set has sizesand is full if it containss(s+1)/2 distinct integers. Kreweras and Loeb suggested the problem of partitioning a run ofms(s+1)/2 integers starting withcintomfull sets of iterated differences of sizes. We show that necessary conditions for this are that 2≤s≤9, and thatmbe sufficiently large in comparison withc. In particular, a single set of iterated differences of sizescontains the integers 1 tos(s+1)/2 (inclusive) iff 2 ≤s≤5. We also discuss connections between this problem and the theory of perfect systems of difference sets
Enumerating Consecutive and Nested Partitions for Graphs

AbstractConsecutive and nested partitions have been extensively studied in the set-partition problem as tools with which to search efficiently for an optimal partition. We extend the study of consecutive and nested partitions on a set of integers to the vertex-set of a graph. A subset of vertices is considered consecutive if the subgraph induced by the subset is connected. In this sense the partition problem on a set of integers can be treated as a special case when the graph is a line. In this paper we give the number of consecutive and nested partitions when the graph is a cycle. We also give a partial order on general graphs with respect to these numbers
On the Capacity of Digraphs

AbstractFor a digraphG=(V,E) letw(Gn) denote the maximum possible cardinality of a subsetSofVnin which for every ordered pair (u1,u2,…,un) and (v1,v2,…,vn) of members ofSthere is some 1≤i≤nsuch that (ui,vi) ∈EThecapacityC(G) ofGisC(G)=limn↦∞[(w(Gn))1/n]. It is shown that for any digraphGwith maximum outdegreed,C(G) ≤d+1. It is also shown that for everynthere is a tournamentTon 2nvertices whose capacity is at leastn, whereas the maximum number of vertices in a transitive subtournament in it is onlyO(logn). This settles a question of Körner and Simonyi
Minimum 0-Extensions of Graph Metrics

AbstractLetH=(T,U) be a connected graph,V⊇Ta set, andca non-negative function on the unordered pairs of elements ofV. In theminimum0-extension problem(*), one is asked to minimize the inner productcmover all metricsmonVsuch that (i)mcoincides with the distance function ofHwithinT; and (ii) eachv∈Vis at zero distance from somes∈T, i.e.m(v,s)=0. This problem is known to be NP-hard ifH=K3(as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable ifH=K2(the minimum cut problem) orH=K2,r(the minimum (2,r)-metric problem). We study problem (*) for all fixedH. More precisely, we consider the linear programming relaxation (**) of (*) that is obtained by dropping condition (ii) above, and callHminimizableif the minima in (*) and (**) coincide for allVandc. Note that for such anHproblem (*) is solvable in strongly polynomial time. Our main theorem asserts thatHis minimizable if and only ifHis bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense thatHcan be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics. Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in [4], we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphsH. Other results are also presented
AnLpVersion of the Beck–Fiala Conjecture

AbstractBeck and Fiala conjectured in 1981 that for any set system S of maximum degreeton a finite ground setX, a coloring χ:X→{−1,+1} exists such that ∣χ(S)∣=O(t) holds for allS∈ S , where χ(S) =∑x∈Sχ(x). We prove a weaker statement, namely that for any fixedp≥ 1, a coloring χ exists such that thepth degree average of ∣χ(S)∣ overS∈S isO(t) . The result also holds if each set is assigned a nonnegative real weight and thepth degree average is taken with these weights (with χ depending on the weights)