pinecone/core-2020-05-10-deduplication huggingface dataset:
Finding Independent Sets inK4-Free 4-Regular Connected Graphs

AbstractLetGbe aK4-free 4-regular connected graph withνvertices. ThenGcontains an independent set of vertices of cardinality at least (7ν−4)/26. Furthermore, the proof will yield a polynomial-time algorithm which will return an independent set of cardinality at least (7ν−4)/26 for any such graph. A family ofK4-free 4-regular connected graphs with maximum independent set of cardinality311νis provided, so that the asymptotically sharp lower bound lies between726and311. We also provide a slight improvement over the best previously known result for triangle-free 5-regular graphs
From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

AbstractThe local adjacency polynomials can be thought of as a generalization, for all graphs, of (the sums of ) the distance polynomials of distance-regular graphs. The term “local” here means that we “see” the graph from a given vertex, and it is the price we must pay for speaking of a kind of distance-regularity when the graph is not regular. It is shown that when the value atλ(the maximum eigenvalue of the graph) of the local adjacency polynomials is large enough, then the eccentricity of the base vertex tends to be small. Moreover, when such a vertex is “tight” (that is, the value of a certain polynomial just fails to satisfy the condition) and fulfils certain additional extremality conditions, then all the polynomials attain their maximum possible values atλ, and the graph turns out to be pseudo-distance-regular around the vertex. As a consequence of the above results, some new characterizations of distance-regular graphs are derived. For example, it is shown that a regular graphΓwithd+1 distinct eigenvalues is distance-regular if, and only if, the number of vertices at distancedfrom any given vertex is the value atλof the highest degree member of an orthogonal system of polynomials, which depend only on the spectrum of the graph
Intersection Representation of Complete Unbalanced Bipartite Graphs

AbstractAp-intersectionrepresentation of a graphGis a map,f, that assigns each vertex a subset of {1,2,…,t} such that {u,v} is an edge if and only if |f(u)∩f(v)|⩾p. The symbolθp(G) denotes this minimumtsuch that ap-intersection representation ofGexists. In 1966 Erdős, Goodman, and Pósa showed that for all graphsGon 2nvertices,θ1(G)⩽θ1(Kn,n)=n2. In 1992, Chung and West conjectured that for all graphsGon 2nvertices,θp(G)⩽θp(Kn,n) whenp⩾1. Subsequently, upper and lower bounds forθp(Kn,n) have been found to be (n2/p)(1+o(1)). We show in this paper that many complete unbalanced bipartite graphs on 2nvertices have a largerp-intersection number thanKn,n. For example, whenp=2,θ2(Kn,n)⩽12n2(1+o(1))<4172n2(1+o(1))⩽θ2(K(5/6)n,(7/6)n)
List Edge and List Total Colourings of Multigraphs

AbstractThis paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B63(1995), 153–158), who proved that the list edge chromatic numberχ′list(G) of a bipartite multigraphGequals its edge chromatic numberχ′(G). It is now proved here that if every edgee=uwof a bipartite multigraphGis assigned a list of at least max{d(u),d(w)} colours, thenGcan be edge-coloured with each edge receiving a colour from its list. If every edgee=uwin an arbitrary multigraphGis assigned a list of at least max{d(u),d(w)}+⌊12min{d(u),d(w)}⌋ colours, then the same holds; in particular, ifGhas maximum degreeΔ=Δ(G) thenχ′list(G)⩽⌊32Δ⌋. Sufficient conditions are given in terms of the maximum degree and maximum average degree ofGin order thatχ′list(G)=Δandχ″list(G)=Δ+1. Consequences are deduced for planar graphs in terms of their maximum degree and girth, and it is also proved that ifGis a simple planar graph andΔ⩾12 thenχ′list(G)=Δandχ″list(G)=Δ+1
Unavoidable Minors of Large 3-Connected Matroids

AbstractThis paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,…,n−1}, the union of every set ofkof these lines has rankk+1
Parallelogram-Free Distance-Regular Graphs

AbstractLetΓ=(X,R) denote a distance-regular graph with distance function ∂ and diameterd⩾4. By a parallelogram of lengthi(2⩽i⩽d), we mean a 4-tuplexyzuof vertices inXsuch that ∂(x,y)=∂(z,u)=1, ∂(x,u)=i, and ∂(x,z)=∂(y,z)=∂(y,u)=i−1. We prove the following theorem.Theorem.LetGamma;denote a distance-regular graph with diameterd⩾4, and intersection numbersa1=0,a2≠0. SupposeΓisQ-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. ThenΓ:has classical parameters (d,b,α,β) withb<−1. By including results in [3], [9], we have the following corollary.Corollary. LetGammadenote a distance-regular graph with theQ-polynomial property. Suppose the diameterd⩾4. Then the following (i)–(ii) are equivalent. (i)Γcontains no parallelograms of any length. (ii) One of the following (iia)–(iic) holds. (iia)Γis bipartite. (iib)Γis a generalized odd graph. (iic)Γhas classical parameters (d,b,α,β) and eitherb<−1 orΓis a Hamming graph or a dual polar graph
Upper Bounds on the Size of Obstructions and Intertwines

AbstractWe give an exponential upper bound inp4on the size of any obstruction for path-width at mostp. We give a doubly exponential upper bound ink5 on the size of any obstruction for tree-width at mostk. We also give an upper bound on the size of any intertwine of two given treesTandT′. The bound is exponential inO(m2logm) wherem⩽max(|V(T)|, |V(T′)|). Finally, we give an upper bound on the size of any intertwine of two given planar graphsHandH′. This bound is triply exponential inO(m5) wherem⩽max(|V(H)|, |V(H′)|). We introduce the concept ofl-length of a family of graphsL. We prove constructively that, if a minor closed familyLhas finitep-length and has obstructions of path-width at mostp, thenLhas a finite number of obstructions. Our proof gives a general upper bound, in termspand thep-length, on the size of any obstruction forL. We obtain a second general upper bound for the case where the obstructions have bounded tree-width. We obtain our upper bounds on obstruction sizes by giving, for the considered familyLand an appropriate integerl, an upper bound on thel-length ofLand applying one of the two general bounds
An Infinite Series of Surfaces with Known 1-Chromatic Number

AbstractThe 1-chromatic numberχ1(S) of a surfaceSis the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. It is proved that if 4n+3 is a prime number,n⩾0, thenχ1(N8(2n+1)2)=R(N8(2n+1)2) whereR(S)=⌊12 (9+81−32E(S))⌋ is Ringel's upper bound forχ1(S),E(S) is the Euler characteristic ofS, andN8(2n+1)2is the nonorientable surface of genus 8(2n+1)2. By Dirichlet's theorem the arithmetic progression 4n+3,n=1, 2, 3, …, contains an infinite number of prime integers. As a result the first known infinite series of surfaces with known 1-chromatic number is obtained
A Vertex-Splitting Lemma, de Werra's Theorem, and Improper List Colourings

AbstractWe prove a new vertex-splitting lemma which states that if a multigraphGhas maximum multiplicity of at mostp, then each vertex u can be split into ⌈(d(u)/p)⌉ new vertices, ⌊(d(u)/p)⌋ of degreep, with the multiple edges being shared out between the new vertices in such a way that each multiple edge remains intact at at least one of its two endpoints. We apply this lemma to deduce very simply the theorem of de Werra that each bipartite multigraph has a balancedk-edge-colouring for each positive integerk. We also apply the lemma to improve two earlier theorems of the first author which state thatχ′ℓs(G)=⌈(Δ(G)/s)⌉ if either (1)Gis a bipartite multigraph, or (2)sis even andGis any multigraph; hereχ′ℓs(G) is thes-improper list chromatic index of G, i.e., the least possible value ofksuch that, if each edge ofGis provided with a list ℓ(e) of at leastkcolours, thenGhas ans-improper edge list colouring (so that at each vertexνnot more than s edges incident withνreceive the same colour), and, on each edge, the colour used lies in the list for that edge. The improvements that we obtain include that for each multiple edge of multiplicitym=mG(u, ν), no colour is used on more than ⌈mG(u, ν)/⌈(Δ(G)/s)⌉⌉+1 edges joininguandν
2-Partition-Transitive Tournaments

AbstractGiven a tournament score sequences1⩾s2⩾…⩾sn, we prove that there exists a tournamentTon vertex set {1, 2, …, n} such that the degree of any vertexiissiand the subtournaments ofTon both the even and the odd vertices are transitive in the given order. This means thatibeatsjwheneveri<jandi≡j(mod2). For any score sequence, we give an algorithm to construct a tournament of the above form, i.e. it is transitive on evens and odds in the given order. This algorithm fixes half of the edges of the tournament and then is similar to the algorithm for constructing a tournament given its score sequence. Another consequence provides asymptotics for the maximum number of edges in score unavoidable digraphs. From a result of Ryser, it is possible to get from any tournament to this special tournament by a sequence of triangle reversals. We show thatn2/2 reversals are always enough and that in some cases (1−o(1))n2/32 are required. We also show that such a sequence of triangle reversals can be found inO(n2) time
Half-Transitive Group Actions on Finite Graphs of Valency 4

AbstractThe action of a subgroupGof automorphisms of a graphXis said to be 12-transitiveif it is vertex- and edge- but not arc-transitive. In this case the graphXis said to be (G, 12)-transitive. In particular,Xis 12-transitiveif it is (AutX, 12)-transitive. The 12-transitive action ofGonXinduces an orientation of the edges ofXwhich is preserved byG. LetXhave valency 4. An even length cycleCinXis aG-alternating cycleif every other vertex ofCis the head and every other vertex ofCis the tail of its two incident edges in the above orientation. It transpires that allG-alternating cycles inXhave the same length and form a decomposition of the edge set ofX(Proposition 2.4); half of this length is denoted byrG(X) and is called theG-radiusofX. Moreover, it is shown that any two adjacentG-alternating cycles ofXintersect in the same number of vertices and that this number, called theG-attachment number aG(X) ofX, divides 2rG(X) (Proposition 2.6). IfXis 12-transitive, we let theradiusand theattachment numberofXbe, respectively, the AutX-radius and the AutX-attachment number ofX. The caseaG(X)=2rG(X) corresponds to the graphXconsisting of twoG-alternating cycles with the same vertex sets and leads to an arc-transitive circulant graph (Proposition 2.4). IfaG(X)=rG(X) we say that the graphXistightly G-attached. In particular, a 12-transitive graphXof valency 4 istightly attachedif it is tightly AutX-attached. A complete classification of tightly attached 12-transitive graphs with odd radius and valency 4 is obtained (Theorem 3.4)
A Nebeský-Type Characterization for Relative Maximum Genus

AbstractThis paper concerns the maximum genus orientable surface upon which a given graph cellularly embeds. Classical theorems of Xuong and Nebeský give exact values for the maximum genus. The former is suited to constructing embeddings while the latter is suited to forbidding embeddings of larger genus. However, using either theorem alone requires an exhaustive search to establish the exact value. Herein we examine relative embeddings of graphs, where certain facial cycles and their orientations have been prescribed. The relative graph analogue of Xuong's theorem is known. In this paper we establish the relative graph analogue of Nebeský's theorem
List Homomorphisms to Reflexive Graphs

AbstractLetHbe a fixed graph. We introduce the following list homomorphism problem: Given an input graphGand for each vertexvofGa “list”L(v)⊆V(H), decide whether or not there is a homomorphismf:G→Hsuch thatf(v)∈L(v) for eachv∈V(G). We discuss this problem primarily in the context of reflexive graphs, i.e., graphs in which each vertex has a loop. We give a polynomial time algorithm to solve the problem whenHis an interval graph and prove that whenHis not an interval graph the problem isNP-complete. If the lists are restricted to induce connected subgraphs ofH, we give a polynomial time algorithm whenHis a chordal graph and prove that whenHis not chordal the problem is againNP-complete. We also argue that the complexity of certain other modifications of the problem (including the retract problem) are likely to be difficult to classify. Finally, we mention some newer results on irreflexive and general graphs
On Packing Connectors

AbstractGiven an undirected graphG=(V,E) and a partition {S,T} ofV, anS−Tconnector is a set of edgesF⊆Esuch that every component of the subgraph (V,F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |δG(V1)∪…∪δG(Vt)|⩾ktfor every collection {V1,…,Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph
Face 2-Colourable Triangular Embeddings of Complete Graphs

AbstractA face 2-colourable triangulation of an orientable surface by a complete graphKnexists if and only ifn≡3 or 7 (mod12). The existence of such triangulations follows from current graph constructions used in the proof of the Heawood conjecture. In this paper we give an alternative construction for half of the residue classn≡7 (mod≡12) which lifts a face 2-colourable triangulation byKmto one byK3m−2. A nonorientable version of this result is discussed as well which enables us to produce nonisomorphic nonorientable triangular embeddings ofKnfor half of the residue classn≡1 (mod6). We also note the existence of nonisomorphic orientable triangular embeddings ofKnforn≡7 (mod12) andn≠7
A Note on Large Graphs of Diameter Two and Given Maximum Degree

AbstractLetvt(d,2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d,2)=d2+1 ford=1,2,3, and 7; for the remaining values ofdwe havevt(d,2)⩽d2−1. The only knowngenerallower bound onvt(d,2), valid forall d, seems to bevt(d,2)⩾⌊(d+2)/2⌋ ⌈(d+2)/2⌉. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d,2)⩾(8/9)(d+12)2for alldof the formd=(3q−1)/2, whereqis a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, ford=7 we obtain as a special case the Hoffman–Singleton graph, and ford=11 andd=13 we have new largest graphs of diameter 2, and degreedon 98 and 162 vertices, respectively
Cycles through Large Degree Vertices in Digraphs: A Generalization of Meyniel's Theorem

AbstractLetD=(V, E) be a digraph with vertex setVof sizenand arc setE. Foru∈V, letd(u) denote the degree ofu. AMeyniel set Mis a subset ofVsuch thatd(u)+d(v)⩾2n−1 for every pair of nonadjacent verticesuandvbelonging toM. In this paper we show that ifDis strongly connected, then every Meyniel setMlies in a (directed) cycle, generalizing Meyniel's theorem. Our proof yields anO(|M|n4) algorithm for finding such a cycle. As a corollary it follows that ifDis strongly connected, thenDcontains a cycle through all vertices of degree ⩾nwhich generalizes a result of Shi for undirected graphs
On Isomorphisms of Connected Cayley Graphs, II

AbstractA Cayley graph or digraph Cay(G, S) of a finite groupGis called aCI-graphofGif, for anyT⊂G, Cay(G, S)≅Cay(G, T) if and only ifSσ=Tfor someσ∈Aut(G). We study the problem of determining which Cayley graphs and digraphs for a given group are CI-graphs. A finite groupGis called aconnected m-DCI-group(orconnected m-CI-group) if all connected Cayley digraphs (or connected Cayley graphs, respectively) ofGof (out)-valency at most m are CI-graphs. For a groupG, letp(G) be the smallest prime divisor of |G|. It was previously shown that all finite groupsGare connected (p(G)−1)-DCI-groups and all finite groupsGare connected 2(p(G)−1)-CI-groups. In this paper, for every primep, we construct infinitely many finite groupsGsuch thatp(G)=pandGis neither a connectedp-DCI-group nor a connected 2p-CI-group, which provides solutions for several open problems in this area
Large Circuits in Binary Matroids of Large Cogirth, I

AbstractLetF7denote the Fano matroid andebe a fixed element ofF7. LetP(F7, e) be the family of matroids obtained by taking the parallel connection of one or more copies ofF7aboute. LetMbe a simple binary matroid such that every cocircuit ofMhas size at leastd⩾3. We show that ifMdoes not have anF7-minor,M≠F*7, andd⩾(r(M)+1)/2 thenMhas a circuit of sizer(M)+1. We also show that ifMis connected,e∈E(M),Mdoes not have both anF7-minor and anF*7-minor, andM∉P(F7, e), thenMhas a circuit that containseand has size at leastd+1
On 2-Connected Spanning Subgraphs with Low Maximum Degree

AbstractGiven a graphG, let ak-trestle ofGbe a 2-connected spanning subgraph ofGof maximum degree at mostk. Also, letχ(G) be the Euler characteristic ofG. This paper shows that every 3-connected graphGhas a (10−2χ(G))-trestle. Ifχ(G)⩽−5, this is improved to 8−2χ(G), and forχ(G)⩽−10, this is further improved to 6−2χ(G). This result is shown to be best possible for almost all values ofχ(G) by the demonstration of 3-connected graphs embedded on each surface of Euler characteristicχ⩽0 which have no (5−2χ)-trestle. Also, it is shown that a 4-connected graph embeddable on a surface with non-negative Euler characteristic has a 3-trestle, approaching a conjecture of Nash-Williams
Even Pairs in Claw-Free Perfect Graphs

AbstractAn even pair in a graph is a pair of non-adjacent vertices such that every chordless path between them has even length. A graph is called strict quasi-parity when every induced subgraph that is not a clique has an even pair, and it is called perfectly contractile when every induced subgraph can be turned into a clique through a sequence of even-pair contractions. In this paper we determine theK1,3-free graphs that are strict quasi-parity and those that are perfectly contractile. We show that for both classes the minimal forbidden configurations are odd holes, antiholes and some line-graphs of bipartite graphs, as conjectured by several authors. Our proofs are constructive and yield polynomial-time algorithms for the recognition of both classes
Classes and Recognition of Curve Contact Graphs

AbstractContact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are introduced and the inclusions between them are described, also the recognition of the contact graphs is studied. As one of the main results, it is proved that the recognition of 3-contact graphs isNP-complete for planar graphs, while the same question for planar triangulations is polynomial
From Hall's Matching Theorem to Optimal Routing on Hypercubes

AbstractWe introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Hall's “marriage” condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(s⩽n) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabin's information disposal algorithm may be improved
Maximal and Minimal Vertex-Critical Graphs of Diameter Two

AbstractA graph is vertex-critical if deleting any vertex increases its diameter. We construct, for eachν⩾5 exceptν=6, a vertex-critical graph of diameter two onνvertices with at least 12ν2−2ν3/2+c1νedges, wherec1is some constant. We show that such a graph contains at most 12ν2−(2/2)ν3/2+c2νedges, wherec2is some constant. We also construct, for eachν⩾5 exceptν=6, a vertex-critical graph of diameter two onνvertices with at most 12 (5ν−12) edges. We show that such a graph must contain at least 12 (5ν−29) edges
The Existence of Exactlym-Coloured Complete Subgraphs

AbstractGiven a graphG, its edges are said to be exactlyx-coloured if we have a surjective map from the edges to some set of colours of sizex. Erickson considered the following statement which he denotedP(c,m): if the edges ofKω—the complete graph on vertex set N—are exactlyc-coloured, then there exists an infinite complete subgraph ofKωwhose edges are exactlym-coloured. Ramsey's Theorem states thatP(c,m) is true form=1 and allc⩾1, and can easily be used to show thatP(c,m) holds whenm=2 andc⩾2. Erickson conjectured thatP(c,m) is false wheneverc>m⩾3. We prove that givenm⩾3 there exists an integerC(m) such thatP(c,m) is false for allc⩾C(m)
On Extremal Connectivity Properties of Unavoidable Matroids

AbstractDing, Oporowski, Oxley, and Vertigan proved that, for alln⩾3, there is an integerN(n) such that every 3-connected matroid with at leastN(n) elements has a minor isomorphic to a wheel or whirl of rankn,M(K3,n) or its dual,U2,n+2or its dual, or a rank-nspike. This paper characterizes each of these classes of unavoidable matroids in terms of an extremal connectivity condition. In particular, it is proved that ifMis a 3-connected matroid of at least rank 7 for which every single-element deletion or contraction is 3-connected but no 2-element deletion or contraction is, thenMis a spike with its tip deleted. It is further proved that ifMis a 3-connected matroid of at least rank 4 for which every single-element deletion is 3-connected but no 1-element contraction or 2-element deletion is, thenM≅M*(K3,n)
Bicycle Dimension and Special Points of the Tutte Polynomial

AbstractFor each pair of algebraic numbers (x,y) and each fieldF, the complexity of computing the Tutte polynomialT(M;x,y) of a matroidMrepresentable overFis determined. This computation is found to be#P-complete except when (x−1)(y−1)=1 or when |F| divides (x−1)(y−1) and (x,y) is one of the seven points (0,−1), (−1,0), (i,−i), (−i,i), (j,j2), (j2,j) or (−1,−1), wherej=e2πi/3. Expressions are given for the Tutte polynomial in the exceptional cases. These expressions involve the bicycle dimension ofMoverF. A related result determines when this bicycle dimension is well defined
An Algorithm for Packing Connectors

AbstractGiven an undirected graphG=(V,E) and a partition {S,T} ofV, anS−Tconnectoris a set of edgesF⊆Esuch that every component of the subgraph (V,F) intersects bothSandT. If eitherSorTis a singleton, then anS−Tconnector is a spanning subgraph ofG. On the other hand, ifGis bipartite with colour classesSandT, then anS−Tconnector is an edge cover ofG(a set of edges covering all vertices). AnS−Tconnector is a common spanning set of two graphic matroids onE. We prove a theorem on packing common spanning sets of certain matroids, generalizing a result of Davies and McDiarmid on strongly base orderable matroids. As a corollary, we obtain anO(τ(n,m)+nm) time algorithm for finding a maximum number ofS−Tconnectors, whereτ(n,m) denotes the complexity of finding a maximum number of edge disjoint spanning trees in a graph onnvertices andmedges. Since the best known bound forτ(n,m) isO(nmlog(m/n)), this bound for packingS−Tconnectors is as good as the current bound for packing spanning trees