Highly Connected Infinite Digraphs Without Edge‐Disjoint Back and Forth Paths Between a Certain Vertex Pair

We construct for all a k‐edge‐connected digraph D with such that there are no edge‐disjoint and paths. We use in our construction “self‐similar” graphs which technique could be useful in other problems as well.     

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 .     

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 .     

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.     

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 .     

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.     

(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.     

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.     

The Game Saturation Number of a Graph

Given a family and a host graph H, a graph is ‐saturated relative to H if no subgraph of G lies in but adding any edge from to G creates such a subgraph. In the ‐saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in , until G becomes ‐saturated relative to H. They aim to maximize or minimize the length of the game, respectively; denotes the length under optimal play (when Max starts). Let denote the family of odd cycles and the family of n‐vertex trees, and write F for when . Our results include , for , for , and for . We also determine ; with , it is n when n is even, m when n is odd and m is even, and when is odd. Finally, we prove the lower bound . The results are very similar when Min plays first, except for the P₄‐saturation game on .     

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.     

On the Existence of Tree Backbones that Realize the Chromatic Number on a Backbone Coloring

A proper k‐coloring of a graph is a function such that , for every . The chromatic number is the minimum k such that there exists a proper k‐coloring of G. Given a spanning subgraph H of G, a q‐backbone k‐coloring of is a proper k‐coloring c of such that , for every edge . The q‐backbone chromatic number is the smallest k for which there exists a q‐backbone k‐coloring of . In this work, we show that every connected graph G has a spanning tree T such that , and that this value is the best possible. As a direct consequence, we get that every connected graph G has a spanning tree T for which , if , or , otherwise. Thus, by applying the Four Color Theorem, we have that every connected nonbipartite planar graph G has a spanning tree T such that . This settles a question by Wang, Bu, Montassier, and Raspaud (J Combin Optim 23(1) (2012), 79–93), and generalizes a number of previous partial results to their question.     

A Hypercube Variant with Small Diameter

This article introduces a new variant of hypercubes . The n‐dimensional twisted hypercube is obtained from two copies of the ‐dimensional twisted hypercube by adding a perfect matching between the vertices of these two copies of . We prove that the n‐dimensional twisted hypercube has diameter . This improves on the previous known variants of hypercube of dimension n and is optimal up to an error of order . Another type of hypercube variant that has similar structure and properties as is also discussed in the last section.     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.     

The 1‐2‐3‐Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex‐coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e. every edge contains at least two vertices with different weighted degrees. In this article, we show that such a weighting is possible from the weight set for all hypergraphs with maximum edge size and not containing edges solely consisting of identical vertices. The number is best possible for this statement.     

Saturation Numbers in Tripartite Graphs

Given graphs H and F, a subgraph is an F‐saturated subgraph of H if , but for all . The saturation number of F in H, denoted , is the minimum number of edges in an F‐saturated subgraph of H. In this article, we study saturation numbers of tripartite graphs in tripartite graphs. For and n₁, n₂, and n₃ sufficiently large, we determine and exactly and within an additive constant. We also include general constructions of ‐saturated subgraphs of with few edges for .     

Extracting List Colorings from Large Independent Sets

We take an application of the Kernel Lemma by Kostochka and Yancey [11] to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore‐degree version of Brooks' Theorem for online list‐coloring. The Ore‐degree of an edge in a graph G is . The Ore‐degree of G is . We show that every graph with and is online ‐choosable. In addition, we prove an upper bound for online list‐coloring triangle‐free graphs: . Finally, we characterize Gallai trees as the connected graphs G with no independent set incident to at least edges.     

Regular Hypertournaments and Arc‐Pancyclicity

A k‐hypertournament H on n vertices () is a pair , where V is the vertex set of H and A is a set of k‐tuples of vertices, called arcs, such that for all subsets with , A contains exactly one permutation of S as an arc. Recently, Li et al. showed that any strong k‐hypertournament H on n vertices, where , is vertex‐pancyclic, an extension of Moon's theorem for tournaments. In this article, we examine several generalizations of regular tournaments and prove the following generalization of Alspach's theorem concerning arc‐pancyclicity: Every Σ‐regular k‐hypertournament on n vertices, where , is arc‐pancyclic.     

Matching Extension Missing Vertices and Edges in Triangulations of Surfaces

Let G be a 5‐connected triangulation of a surface Σ different from the sphere, and let be the Euler characteristic of Σ. Suppose that with even and M and N are two matchings in of sizes m and n respectively such that . It is shown that if the pairwise distance between any two elements of is at least five and the face‐width of the embedding of G in Σ is at least , then there is a perfect matching M₀ in containing M such that .     

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 .