Improper Coloring of Sparse Graphs with a Given Girth,II: Constructions

A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all .     

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.     

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 .     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

Extremal Graphs With a Given Number of Perfect Matchings

Let denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that for some constant when n is at least some constant . For , they also determined and . For fixed p, we show that the extremal graphs for all n are determined by those with vertices. As a corollary, a computer search determines and for . We also present lower bounds on proving that for (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on . Our structural results are based on Lovász's Cathedral Theorem.     

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 .     

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 , .     

The Set of Circular Flow Numbers of Regular Graphs

This article determines the set of the circular flow numbers of regular graphs. Let be the set of the circular flow numbers of graphs, and be the set of the circular flow numbers of d‐regular graphs. If d is even, then . For it is known 6 that . We show that . Hence, the interval is the only gap for circular flow numbers of ‐regular graphs between and 5. Furthermore, if Tutte's 5‐flow conjecture is false, then it follows, that gaps for circular flow numbers of graphs in the interval [5, 6] are due for all graphs not just for regular graphs.     

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.     

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.     

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.     

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 .     

Edmonds' Branching Theorem in Digraphs Without Forward‐Infinite Paths

Let D be a finite digraph, and let be nonempty subsets of . The (strong form of) Edmonds' branching theorem states that there are pairwise edge‐disjoint spanning branchings in D such that the root set of is if and only if for all the number of ingoing edges of X is greater than or equal to the number of sets disjoint from X. As was shown by R. Aharoni and C. Thomassen (J Graph Theory 13 (1989), 71–74), this theorem does not remain true for infinite digraphs. Thomassen also proved that for the class of digraphs without backward‐infinite paths, the above theorem of Edmonds remains true. Our main result is that for digraphs without forward‐infinite paths, Edmonds' branching theorem remains true as well.     

A Combined Logarithmic Bound on the Chromatic Index of Multigraphs

For a multigraph G, the integer round‐up of the fractional chromatic index provides a good general lower bound for the chromatic index . For an upper bound, Kahn 1996 showed that for any real there exists a positive integer N so that whenever . We show that for any multigraph G with order n and at least one edge, ). This gives the following natural generalization of Kahn's result: for any positive reals , there exists a positive integer N so that + c whenever . We also compare the upper bound found here to other leading upper bounds.     

An Improved Bound for Vertex Partitions by Connected Monochromatic K‐Regular Graphs

Improving a result of Sárközy and Selkow, we show that for all integers there exists a constant such that if and the edges of the complete graph are colored with r colors then the vertex set of can be partitioned into at most vertex disjoint connected monochromatic k‐regular subgraphs and vertices. This is close to best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 127–145, 2013     

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.     

The Degree‐Diameter Problem for Claw‐Free Graphs and Hypergraphs

We study the degree‐diameter problem for claw‐free graphs and 2‐regular hypergraphs. Let be the largest order of a claw‐free graph of maximum degree Δ and diameter D. We show that , where , for any D and any even . So for claw‐free graphs, the well‐known Moore bound can be strengthened considerably. We further show that for with (mod 4). We also give an upper bound on the order of ‐free graphs of given maximum degree and diameter for . We prove similar results for the hypergraph version of the degree‐diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most . For 2‐regular hypergraph of rank and any diameter D, we improve this bound to , where . Our construction of claw‐free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank .     

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.