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.     

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     

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.     

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.     

On Hypohamiltonian and Almost Hypohamiltonian Graphs

A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that is non‐hamiltonian, and for any vertex the graph is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every . During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.     

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 .     

Total Domination in Graphs with Diameter 2

The total domination number of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, . This bound is optimal in the sense that given any , there exist graphs G with diameter 2 of all sufficiently large even orders n such that .     

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 .     

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 .     

Extremal Graphs for Homomorphisms II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize . We prove that if H is loop‐threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges that maximizes . Similarly, we show that loop‐quasi‐threshold image graphs have quasi‐threshold extremal graphs. In the case , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for . Also in this article, using similar techniques, we determine a set of extremal graphs for “the fox,” a graph formed by deleting the loop on one of the end‐vertices of . The fox is the unique connected loop‐threshold image graph on at most three vertices for which the extremal problem was not previously solved.     

Total Vertex Irregularity Strength of Dense Graphs

Consider a graph of minimum degree δ and order n. Its total vertex irregularity strength is the smallest integer k for which one can find a weighting such that for every pair of vertices of G. We prove that the total vertex irregularity strength of graphs with is bounded from above by . One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.     

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.     

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A ‐list assignment L of a graph G is a function that assigns to each vertex v a list of at least k colors and for any adjacent pair , the lists and share at most d colors. A graph G is ‐choosable if there exists an L‐coloring of G for every ‐list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4‐cycles are (3, 1)‐choosable and that planar graphs without 5‐ and 6‐cycles are (3, 1)‐choosable. In addition, we give an alternative and slightly stronger proof that triangle‐free planar graphs are (3, 1)‐choosable.     

Cycles Avoiding a Color in Colorful Graphs

The Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .     

(1, k)‐Coloring of Graphs with Girth at Least Five on a Surface

A graph is ‐colorable if its vertex set can be partitioned into r sets so that the maximum degree of the graph induced by is at most for each . For a given pair , the question of determining the minimum such that planar graphs with girth at least g are ‐colorable has attracted much interest. The finiteness of was known for all cases except when . Montassier and Ochem explicitly asked if d₂(5, 1) is finite. We answer this question in the affirmative with ; namely, we prove that all planar graphs with girth at least five are (1, 10)‐colorable. Moreover, our proof extends to the statement that for any surface S of Euler genus γ, there exists a where graphs with girth at least five that are embeddable on S are (1, K)‐colorable. On the other hand, there is no finite k where planar graphs (and thus embeddable on any surface) with girth at least five are (0, k)‐colorable.     

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Galvin showed that for all fixed δ and sufficiently large n, the n‐vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n‐vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n‐vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.     

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.     

List Colorings with Distinct List Sizes,the Case of Complete Bipartite Graphs

Let be a function on the vertex set of the graph . The graph G is f‐choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, , is the minimum of , over all functions f such that G is f‐choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1–33, Random Struct Algor 16 (2000), 364–368) that if G has average degree d, then the usual choice number is at least , so they grow simultaneously. In this article, we show that can be bounded while the minimum degree . Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph .     

List‐Coloring the Squares of Planar Graphs without 4‐Cycles and 5‐Cycles

Let G be a planar graph without 4‐cycles and 5‐cycles and with maximum degree . We prove that . For arbitrarily large maximum degree Δ, there exist planar graphs of girth 6 with . Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list‐coloring. In addition, we prove bounds for ‐labeling. Specifically, and, more generally, , for positive integers p and q with . Again, these bounds come from a greedy coloring, so they immediately extend to the list‐coloring and online list‐coloring variants of this problem.