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1.
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 . 相似文献
2.
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 . 相似文献
3.
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. 相似文献
4.
Vladimir P. Korzhik 《Journal of Graph Theory》2013,74(2):133-142
We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ). 相似文献
5.
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 . 相似文献
6.
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 . 相似文献
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8.
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. 相似文献
9.
We prove a conjecture of Ohba that says that every graph G on at most vertices satisfies . 相似文献
10.
Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley–Wilf limits. We investigate the maximum number of edges in ‐interval minor‐free bipartite graphs. We determine exact values when and describe the extremal graphs. For , lower and upper bounds are given and the structure of ‐interval minor‐free graphs is studied. 相似文献
11.
Stephen G. Hartke Derrick Stolee Douglas B. West Matthew Yancey 《Journal of Graph Theory》2013,73(4):449-468
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. 相似文献
12.
Peng‐An Chen 《Journal of Graph Theory》2015,79(3):233-248
For positive integers n and s, a subset [n] is s‐stable if for distinct . The s‐stable r‐uniform Kneser hypergraph is the r‐uniform hypergraph that has the collection of all s‐stable k‐element subsets of [n] as vertex set and whose edges are formed by the r‐tuples of disjoint s‐stable k‐element subsets of [n]. Meunier ( 21 ) conjectured that for positive integers with , and , the chromatic number of s‐stable r ‐uniform Kneser hypergraphs is equal to . It is a generalized version of the conjecture proposed by Alon et al. ( 1 ). Alon et al. ( 1 ) confirmed Meunier's conjecture for with arbitrary positive integer q. Lin et al. ( 17 ) studied the kth chromatic number of the Mycielskian of the ordinary Kneser graphs for . They conjectured that for . The case was proved by Mycielski ( 22 ). Lin et al. ( 17 ) confirmed their conjecture for , or when n is a multiple of k or . In this paper, we investigate the multichromatic number of the usual s ‐stable Kneser graphs . With the help of Fan's (1952) combinatorial lemma, we show that Meunier's conjecture is true for r is a power of 2 and s is a multiple of r, and Lin‐Liu‐Zhu's conjecture is true for . 相似文献
13.
Consider a simple graph and its proper edge coloring c with the elements of the set . We say that c is neighbor set distinguishing (or adjacent strong) if for every edge , the set of colors incident with u is distinct from the set of colors incident with v. Let us then consider a stronger requirement and suppose we wish to distinguishing adjacent vertices by sums of their incident colors. In both problems the challenging conjectures presume that such colorings exist for any graph G containing no isolated edges if only . We prove that in both problems is sufficient. The proof is based on the Combinatorial Nullstellensatz, applied in the “sum environment.” In fact the identical bound also holds if we use any set of k real numbers instead of as edge colors, and the same is true in list versions of the both concepts. In particular, we therefore obtain that lists of length ( in fact) are sufficient for planar graphs. 相似文献
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15.
Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph
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Thomas A. McCourt 《Journal of Graph Theory》2014,77(1):68-87
We construct a face two‐colourable, blue and green say, embedding of the complete graph in a nonorientable surface in which there are blue faces each of which have a hamilton cycle as their facial walk and green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of , for all and . Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph , for where and . 相似文献
16.
Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles,Matchings, or Trees
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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. 相似文献
17.
Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications
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In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for . 相似文献
18.
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. 相似文献
19.
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 . 相似文献
20.
This article intends to study some functors from the category of graphs to itself such that, for any graph G, the circular chromatic number of is determined by that of G. In this regard, we investigate some coloring properties of graph powers. We show that provided that . As a consequence, we show that if , then . In particular, and has no subgraph with circular chromatic number equal to . This provides a negative answer to a question asked in (X. Zhu, Discrete Math, 229(1–3) (2001), 371–410). Moreover, we investigate the nth multichromatic number of subdivision graphs. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that . 相似文献