Vizing's 2‐Factor Conjecture Involving Large Maximum Degree

Let G be an n‐vertex simple graph, and let and denote the maximum degree and chromatic index of G, respectively. Vizing proved that or . Define G to be Δ‐critical if and for every proper subgraph H of G. In 1965, Vizing conjectured that if G is an n‐vertex Δ‐critical graph, then G has a 2‐factor. Luo and Zhao showed if G is an n‐vertex Δ‐critical graph with , then G has a hamiltonian cycle, and so G has a 2‐factor. In this article, we show that if G is an n‐vertex Δ‐critical graph with , then G has a 2‐factor.     

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Let G be a simple undirected connected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a proper Δ‐coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k‐coloring, , a Δ‐coloring of G can be obtained by a sequence of recolorings using only the original k colors unless

• – G is a complete graph or a cycle with an odd number of vertices, or
• – , G is Δ‐regular and, for each vertex v in G, no two neighbors of v are colored alike.

We use this result to study the reconfiguration graph of the k‐colorings of G. The vertex set of is the set of all possible k‐colorings of G and two colorings are adjacent if they differ on exactly one vertex. We prove that for , consists of isolated vertices and at most one further component that has diameter . This result enables us to complete both a structural and an algorithmic characterization for reconfigurations of colorings of graphs of bounded maximum degree.     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

Large ‐ or ‐Minors in 3‐Connected Graphs

There are numerous results bounding the circumference of certain 3‐connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3‐connected graph, however. Oporowski, Oxley, and Thomas (J Combin Theory Ser B 57 (1993), 2, 239–257) proved the following result in 1993. For every positive integer k, there is an integer such that every 3‐connected graph with at least n vertices contains a ‐ or ‐minor. This result implies that the size of the largest bond in a 3‐connected graph grows with the order of the graph. Oporowski et al. obtained a huge function iteratively. In this article, we first improve the above authors' result and provide a significantly smaller and simpler function . We then use the result to obtain a lower bound for the largest bond of a 3‐connected graph by showing that any 3‐connected graph on n vertices has a bond of size at least . In addition, we show the following: Let G be a 3‐connected planar or cubic graph on n vertices. Then for any , G has a ‐minor with , and thus a bond of size at least .     

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.     

Cycle‐Saturated Graphs with Minimum Number of Edges

A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by . We prove holds for all , where is a cycle with length k. A graph G is H‐semisaturated if contains more copies of H than G does for . Let be the minimum size of an n‐vertex H‐semisaturated graph. We have We conjecture that our constructions are optimal for . © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013     

Counterexamples to the List Square Coloring Conjecture

The square G² of a graph G is the graph defined on such that two vertices u and v are adjacent in G² if the distance between u and v in G is at most 2. Let and be the chromatic number and the list chromatic number of a graph H, respectively. A graph H is called chromatic‐choosable if . It is an interesting problem to find graphs that are chromatic‐choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that for every graph G, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value can be arbitrarily large.     

Constructing a Family of 4‐Critical Planar Graphs with High Edge Density

A graph is a k‐critical graph if G is not ‐colorable but every proper subgraph of G is ‐colorable. In this article, we construct a family of 4‐critical planar graphs with n vertices and edges. As a consequence, this improves the bound for the maximum edge density attained by Abbott and Zhou. We conjecture that this is the largest edge density for a 4‐critical planar graph.     

Subcubic Edge‐Chromatic Critical Graphs Have Many Edges

We consider graphs G with such that and for every edge e, so‐called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, , is such a graph and has average degree only . He showed that every critical graph has average degree at least , and asked if is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if G is a subcubic critical graph other than , then G has average degree at least . This bound is best possible, as shown by the Hajós join of two copies of .     

A Closure for 1‐Hamilton‐Connectedness in Claw‐Free Graphs

A graph G is 1‐Hamilton‐connected if is Hamilton‐connected for every vertex . In the article, we introduce a closure concept for 1‐Hamilton‐connectedness in claw‐free graphs. If is a (new) closure of a claw‐free graph G, then is 1‐Hamilton‐connected if and only if G is 1‐Hamilton‐connected, is the line graph of a multigraph, and for some , is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassen's Conjecture (every 4‐connected line graph is hamiltonian) is equivalent to the statement that every 4‐connected claw‐free graph is 1‐Hamilton‐connected, and we present results showing that every 5‐connected claw‐free graph with minimum degree at least 6 is 1‐Hamilton‐connected and that every 4‐connected claw‐free and hourglass‐free graph is 1‐Hamilton‐connected.     

Petersen Cores and the Oddness of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. We study lists by three 1‐factors, and call with a ‐core of G. If G is not 3‐edge‐colorable, then . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that if , then is an upper bound for the girth of G. We show that bounds the oddness of G as well. We prove that . If , then every ‐core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4‐edge‐connected cubic graph G with . On the other hand, the difference between and can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer , there exists a bridgeless cubic graph G such that .     

Partial Online List Coloring of Graphs

For a graph G, let be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s‐choosable and , then . In this article, we consider the online version of this conjecture. Let be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s‐choosable and then . This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers , . As a consequence, if s is a multiple of t, then . We also prove that if G is online s‐choosable and , then and for any , .     

Face 2‐Colorable Embeddings with Faces of Specified Lengths

Suppose and are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2‐colorable planar embedding in which the faces of one color have boundary lengths and the faces of the other color have boundary lengths . Such a graph is said to have a planar ‐biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths and also into t cycles of lengths . Such a graph is said to be ‐decomposable.     

Average Degree in Graph Powers

The kth power of a simple graph G, denoted by , is the graph with vertex set where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of . Here we prove that if G is connected with minimum degree and , then G⁴ has average degree at least . We also prove that if G is a connected d‐regular graph on n vertices with diameter at least , then the average degree of is at least Both these results are shown to be essentially best possible; the second is best possible even when is arbitrarily large.     

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.     

Circular Chromatic Indices of Regular Graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are adjacent. It is known that for any , there is a 3‐regular simple graph G with . This article proves the following results: Assume is an odd integer. For any , there is an n‐regular simple graph G with . For any , there is an n‐regular multigraph G with .     

Unique Colorability and Clique Minors

For a graph G, let denote the largest k such that G has k pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let denote the largest k such that G has k pairwise disjoint pairwise adjacent connected subgraphs of size 1 or 2. Hadwiger 's conjecture states that , where is the chromatic number of G. Seymour conjectured for all graphs without antitriangles, that is,  three pairwise nonadjacent vertices. Here we concentrate on graphs G with exactly one ‐coloring. We prove generalizations of the following statements: (i) if and G has exactly one ‐coloring then , where the proof does not use the four‐color‐theorem, and (ii) if G has no antitriangles and G has exactly one ‐coloring then .     

Tree‐Chromatic Number Is Not Equal to Path‐Chromatic Number*

For a graph G and a tree‐decomposition of G, the chromatic number of is the maximum of , taken over all bags . The tree‐chromatic number of G is the minimum chromatic number of all tree‐decompositions of G. The path‐chromatic number of G is defined analogously. In this article, we introduce an operation that always increases the path‐chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path‐chromatic number and tree‐chromatic number are different. This settles a question of Seymour (J Combin Theory Ser B 116 (2016), 229–237). Our results also imply that the path‐chromatic numbers of the Mycielski graphs are unbounded.     

Eigenvalues of ‐Free Graphs and the Connectivity of Their Independence Complexes

Let G be a graph on n vertices, with maximal degree d, and not containing as an induced subgraph. We prove:

• 1.
• 2.

Here is the maximal eigenvalue of the Laplacian of G, is the independence complex of G, and denotes the topological connectivity of a complex plus 2. These results provide improved bounds for the existence of independent transversals in ‐free graphs.