Exhaustive generation of k‐critical ‐free graphs

We describe an algorithm for generating all k‐critical ‐free graphs, based on a method of Hoàng et al. (A graph G is k‐critical H‐free if G is H‐free, k‐chromatic, and every H‐free proper subgraph of G is ‐colorable). Using this algorithm, we prove that there are only finitely many 4‐critical ‐free graphs, for both and . We also show that there are only finitely many 4‐critical ‐free graphs. For each of these cases we also give the complete lists of critical graphs and vertex‐critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3‐colorability problem in the respective classes. In addition, we prove a number of characterizations for 4‐critical H‐free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4‐critical planar ‐free graphs is finite. We also determine all 52 4‐critical planar P₇‐free graphs. We also prove that every P₁₁‐free graph of girth at least five is 3‐colorable, and show that this is best possible by determining the smallest 4‐chromatic P₁₂‐free graph of girth at least five. Moreover, we show that every P₁₄‐free graph of girth at least six and every P₁₇‐free graph of girth at least seven is 3‐colorable. This strengthens results of Golovach et al.     

Amalgams and χ‐Boundedness

A class of graphs is hereditary if it is closed under isomorphism and induced subgraphs. A class of graphs is χ‐bounded if there exists a function such that for all graphs , and all induced subgraphs H of G, we have that . We prove that proper homogeneous sets, clique‐cutsets, and amalgams together preserve χ‐boundedness. More precisely, we show that if and are hereditary classes of graphs such that is χ‐bounded, and such that every graph in either belongs to or admits a proper homogeneous set, a clique‐cutset, or an amalgam, then the class is χ‐bounded. This generalizes a result of [J Combin Theory Ser B 103(5) (2013), 567–586], which states that proper homogeneous sets and clique‐cutsets together preserve χ‐boundedness, as well as a result of [European J Combin 33(4) (2012), 679–683], which states that 1‐joins preserve χ‐boundedness. The house is the complement of the four‐edge path. As an application of our result and of the decomposition theorem for “cap‐free” graphs from [J Graph Theory 30(4) (1999), 289–308], we obtain that if G is a graph that does not contain any subdivision of the house as an induced subgraph, then .     

Chordal 2‐Connected Graphs and Spanning Trees

We present a transformation on a chordal 2‐connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with , the threshold graph having n vertices and m edges that consists of an ‐clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2‐connected chordal graphs on n vertices and m edges.     

On the structure of (pan,even hole)‐free graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph that consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)‐free graph can be decomposed by clique cutsets into essentially unit circular‐arc graphs. This structure theorem is the basis of our ‐time certifying algorithm for recognizing (pan, even hole)‐free graphs and for our ‐time algorithm to optimally color them. Using this structure theorem, we show that the tree‐width of a (pan, even hole)‐free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 time the clique number.     

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

A graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. ( http://arxiv.org/abs/1308.6444 ).     

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1‐regular subgraph. It is known that every n‐vertex graph has at most  maximal induced matchings, and this bound is the best possible. We prove that every n‐vertex triangle‐free graph has at most  maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K_{3, 3}. Our result implies that all maximal induced matchings in an n‐vertex triangle‐free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle‐free graph.     

Coloring (gem, co‐gem)‐free graphs

A gem is a graph that consists of a path on four vertices plus a vertex adjacent to all four vertices of the path. A co‐gem is the complement of a gem. We prove that every (gem, co‐gem)‐free graph G satisfies the inequality (a special case of a conjecture of Gyárfás) and the inequality (a special case of a conjecture of Reed). Moreover, we give an ‐time algorithm that computes the chromatic number of any (gem, co‐gem)‐free graph with n vertices, while the existing algorithm in the literature takes .     

Forbidden Pairs of Disconnected Graphs Implying Hamiltonicity

Let H be a given graph. A graph G is said to be H‐free if G contains no induced copies of H. For a class of graphs, the graph G is ‐free if G is H‐free for every . Bedrossian characterized all the pairs of connected subgraphs such that every 2‐connected ‐free graph is hamiltonian. Faudree and Gould extended Bedrossian's result by proving the necessity part of the result based on infinite families of non‐hamiltonian graphs. In this article, we characterize all pairs of (not necessarily connected) graphs such that there exists an integer n₀ such that every 2‐connected ‐free graph of order at least n₀ is hamiltonian.     

Leaf‐Critical and Leaf‐Stable Graphs

The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property for each or for each . These graphs will be called ‐leaf‐critical and l‐leaf‐stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3‐leaf‐critical graphs (that turn out to be the so‐called hypotraceable graphs) was an open problem until 1975. We show that l‐leaf‐stable and l‐leaf‐critical graphs exist for every integer , moreover for n sufficiently large, planar l‐leaf‐stable and l‐leaf‐critical graphs exist on n vertices. We also characterize 2‐fragments of leaf‐critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf‐critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.     

Applications of Hajós‐Type Constructions to the Hedetniemi Conjecture

Hedetniemi conjectured in 1966 that if G and H are finite graphs with chromatic number n, then the chromatic number of the direct product of G and H is also n. We mention two well‐known results pertaining to this conjecture and offer an improvement of the one, which partially proves the other. The first of these two results is due to Burr et al. (Ars Combin 1 (1976), 167–190), who showed that when every vertex of a graph G with is contained in an n‐clique, then whenever . The second, by Duffus et al. (J Graph Theory 9 (1985), 487–495), and, obtained independently by Welzl (J Combin Theory Ser B 37 (1984), 235–244), states that the same is true when G and H are connected graphs each with clique number n. Our main result reads as follows: If G is a graph with and has the property that the subgraph of G induced by those vertices of G that are not contained in an n‐clique is homomorphic to an ‐critical graph H, then . This result is an improvement of the result by the first authors. In addition we will show that our main result implies a special case of the result by the second set of authors. Our approach will employ a construction of a graph F, with chromatic number , that is homomorphic to G and H.     

Retracts of Products of Chordal Graphs

In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K_{2, 3}, the 4‐wheel minus one spoke , and the k‐wheels (for as induced subgraphs. We also show that these graphs G are exactly the cage‐amalgamation graphs as introduced by Bre?ar and Tepeh Horvat (Cage‐amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071–1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed with an intrinsic Euclidean metric, is a CAT(0) space. This generalizes similar results about median graphs as retracts of hypercubes (products of edges) and median graphs as 1‐skeletons of CAT(0) cubical complexes. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 161–180, 2013     

Toroidal graphs containing neither nor 6‐cycles are 4‐choosable

The choosability of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that , and if and only if G contains K₇. Cai et al. (J Graph Theory 65(1) (2010), 1–15) proved that a toroidal graph G without 7‐cycles is 6‐choosable, and if and only if G contains K₆. They also proved that a toroidal graph G without 6‐cycles is 5‐choosable, and conjectured that if and only if G contains K₅. We disprove this conjecture by constructing an infinite family of non‐4‐colorable toroidal graphs with neither K₅ nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither (a K₅ missing one edge) nor 6‐cycles are 4‐choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4‐choosable.     

Local Clique Covering of Claw‐Free Graphs

A clique covering of a simple graph G is a collection of cliques of G covering all the edges of G such that each vertex is contained in at most k cliques. The smallest k for which G admits a clique covering is called the local clique cover number of G and is denoted by lcc(G). Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well‐known problems are discussed. In particular, it is proved that the local clique cover number of every claw‐free graph is at most , where Δ is the maximum degree of the graph and c is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen et al. (Clique covering the edges of a locally cobipartite graph, Discrete Math 219(1–3)(2000), 17–26), we prove that the clique cover number of every connected claw‐free graph on n vertices with the minimum degree δ, is at most , where c is a constant.     

Minimum Degrees of Minimal Ramsey Graphs for Almost‐Cliques

For graphs F and H, we say F is Ramsey for H if every 2‐coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H‐minimal if F is Ramsey for H and there is no proper subgraph of F so that is Ramsey for H. Burr et al. defined to be the minimum degree of F over all Ramsey H‐minimal graphs F. Define to be a graph on vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that for all values ; it was previously known that , so it is surprising that is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that for all graphs H, where is the minimum degree of H; Szabó et al. investigated which graphs have this property and conjectured that all bipartite graphs H without isolated vertices satisfy . Fox et al. further conjectured that all connected triangle‐free graphs with at least two vertices satisfy this property. We show that d‐regular 3‐connected triangle‐free graphs H, with one extra technical constraint, satisfy ; the extra constraint is that H has a vertex v so that if one removes v and its neighborhood from H, the remainder is connected.     

Uniquely Cycle‐Saturated Graphs

Given a graph F, a graph G is uniquely F‐saturated if F is not a subgraph of G and adding any edge of the complement to G completes exactly one copy of F. In this article, we study uniquely ‐saturated graphs. We prove the following: (1) a graph is uniquely C₅‐saturated if and only if it is a friendship graph. (2) There are no uniquely C₆‐saturated graphs or uniquely C₇‐saturated graphs. (3) For , there are only finitely many uniquely ‐saturated graphs (we conjecture that in fact there are none). Additionally, our results show that there are finitely many k‐friendship graphs (as defined by Kotzig) for .     

Maximizing H‐Colorings of a Regular Graph

For graphs G and H, a homomorphism from G to H, or H‐coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H‐colorings admitted by an n‐vertex, d‐regular graph, for each H. Specifically, writing for the number of H‐colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n‐vertex, d‐regular, loopless graph G, we have where is the complete bipartite graph with d vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of H. This allows us to characterize those H for which is eventually (for all sufficiently large d) larger than and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d‐regular G, we have where as . More precise results are obtained in some special cases.     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

Monochromatic Kr‐Decompositions of Graphs

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic H‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let be the smallest number ? such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H‐decomposition with at most ? parts. Here, we study the function for and .     

Topological Minors of Cover Graphs and Dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that ‐free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.     

Existence of u‐Representation of Graphs

Recently, Jones et al. (Electron J Comb 22(2) (2015), #P2.53) introduced the study of u‐representable graphs, where u is a word over containing at least one 1. The notion of a u‐representable graph is a far‐reaching generalization of the notion of a word‐representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is 11???1‐representable assuming that the number of 1s is at least three, while the class of 12‐representable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word‐representable graphs corresponding to 11‐representable graphs. Further studies in this direction were conducted by Nabawanda (M.Sc. thesis, 2015), who has shown, in particular, that the class of 112‐representable graphs is not included in the class of word‐representable graphs. Jones et al. raised a question on classification of u‐representable graphs at least for small values of u . In this article, we show that if u is of length at least 3 then any graph is u‐representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting nonequivalent cases in the theory of u‐representable graphs, namely, those of and .