Structure Theorem for U5‐free Tournaments

Let U₅ be the tournament with vertices v₁, …, v₅ such that , and if , and . In this article, we describe the tournaments that do not have U₅ as a subtournament. Specifically, we show that if a tournament G is “prime”—that is, if there is no subset , , such that for all , either for all or for all —then G is U₅‐free if and only if either G is a specific tournament or can be partitioned into sets X, Y, Z such that , , and are transitive. From the prime U₅‐free tournaments we can construct all the U₅‐free tournaments. We use the theorem to show that every U₅‐free tournament with n vertices has a transitive subtournament with at least vertices, and that this bound is tight.     

Minors in ‐ Chromatic Graphs,II

Let denote the graph obtained from the complete graph by deleting the edges of some ‐subgraph. The author proved earlier that for each fixed s and , every graph with chromatic number has a minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for .     

Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles,Matchings, or Trees

We seek the maximum number of colors in an edge‐coloring of the complete graph not having t edge‐disjoint rainbow spanning subgraphs of specified types. Let , , and denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove for and for . We prove for and for . We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than q edges at a vertex.     

Minimum Vertex Degree Threshold for ‐tiling*

We prove that the vertex degree threshold for tiling (the 3‐uniform hypergraph with four vertices and two triples) in a 3‐uniform hypergraph on vertices is , where if and otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.     

On the Edit Distance from ‐Free Graphs

The edit distance between two graphs on the same vertex set is defined to be the size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, , is a function of p, and measures, asymptotically, the furthest graph of edge density p from under this metric. In this article, we address the hereditary property , the property of having no induced copy of the complete bipartite graph with two vertices in one class and t in the other. Employing an assortment of techniques and colored regularity graph constructions, we are able to determine the edit distance function over the entire domain when and extend the interval over which the edit distance function for is known for all values of t, determining its maximum value for all odd t. We also prove that the function for odd t has a nontrivial interval on which it achieves its maximum. These are the only known principal hereditary properties for which this occurs. In the process of studying this class of functions, we encounter some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz.     

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 .     

Two‐sided Group Digraphs and Graphs

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.     

On 2‐Colorings of Hypergraphs

In this article, we continue the study of 2‐colorings in hypergraphs. A hypergraph is 2‐colorable if there is a 2‐coloring of the vertices with no monochromatic hyperedge. Let denote the class of all k‐uniform k‐regular hypergraphs. It is known (see Alon and Bregman [Graphs Combin. 4 (1988) 303–306] and Thomassen [J. Amer. Math. Soc. 5 (1992), 217–229] that every hypergraph is 2‐colorable, provided . As remarked by Alon and Bregman the result is not true when , as may be seen by considering the Fano plane. Indeed there are several constructions for building infinite families of hypergraphs in that are not 2‐colorable. Our main result in this paper is a strengthening of the above results. For this purpose, we define a set X of vertices in a hypergraph H to be a free set in H if we can 2‐color such that every edge in H receives at least one vertex of each color. We conjecture that for , every hypergraph has a free set of size in H. We show that the bound cannot be improved for any and we prove our conjecture when . Our proofs use results from areas such as transversal in hypergraphs, cycles in digraphs, and probabilistic arguments.     

Spanning Trees with Vertices Having Large Degrees

Let G be a connected simple graph, and let f be a mapping from to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least . We show that if for any nonempty subset , then a connected graph G has a spanning tree such that for all , where is the set of neighbors v of vertices in S with , , and is the degree of x in T. This is an improvement of several results, and the condition is best possible.     

A Note on Even Cycles and Quasirandom Tournaments

A cycle in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from to . In this short article, we show that for every fixed even integer , if close to half of the k‐cycles in a tournament T are even, then T must be quasirandom.This resolves an open question raised in 1991 by Chung and Graham 1991.     

Finding for a Surface Σ of Characteristic −4

For each surface Σ, we define max G is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .     

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.     

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.     

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.     

2‐Distance Coloring of Sparse Graphs

For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.     

Approximations to m‐Colored Complete Infinite Hypergraphs

Given an edge coloring of a graph with a set of m colors, we say that the graph is exactly m‐colored if each of the colors is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number and for all sufficiently large k, there is a k‐coloring of the complete graph on such that no complete infinite subgraph is exactly m‐colored. In the light of this result, we consider the question of how close we can come to finding an exactly m‐colored complete infinite subgraph. We show that for a natural number m and any finite coloring of the edges of the complete graph on with m or more colors, there is an exactly ‐colored complete infinite subgraph for some satisfying ; this is best possible up to the additive constant. We also obtain analogous results for this problem in the setting of r‐uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalizations of this conjecture to r‐uniform hypergraphs.     

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 .     

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let denote the number of convex cycles of a simple graph G of order n, size m, and girth . It is proved that and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.     

Finite 2‐Geodesic Transitive Graphs of Prime Valency

We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.