Search for: All records

Abstract Let be a simple graph. Let and be the maximum degree and the chromatic index of , respectively. We calloverfullif , andcriticalif for every proper subgraph of . Clearly, if is overfull then . Thecoreof , denoted by , is the subgraph of induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer , if is critical with and , then is overfull. 

Abstract Let be a simple graph and be the chromatic index of . We call a ‐critical graphif for every edge of , where is maximum degree of . Let be an edge of ‐critical graph and be an (proper) edge ‐coloring of . Ane‐fanis a sequence of alternating vertices and distinct edges such that edge is incident with or , is another endvertex of and is missing at a vertex before for each with . In this paper, we prove that if , where and denote the degrees of vertices and , respectively, then colors missing at different vertices of are distinct. Clearly, a Vizing fan is an ‐fan with the restricting that all edges being incident with one fixed endvertex of edge . This result gives a common generalization of several recently developed new results on multifan, double fan, Kierstead path of four vertices, and broom. By treating some colors of edges incident with vertices of low degrees as missing colors, Kostochka and Stiebitz introduced ‐fan. In this paper, we also generalize the ‐fan from centered at one vertex to one edge. 

Abstract A graph is said to be ‐free if it does not contain any subdivision of as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ‐free graph is 4‐colorable. In this paper, we show that this conjecture is true for the class of {, diamond, bowtie}‐free graphs, where a diamond is the graph obtained from by removing one edge and a bowtie is the graph consisting of two triangles with one vertex identified. 

Abstract Let be a multigraph with maximum degree and chromatic index . If is bipartite then . Otherwise, by a theorem of Goldberg, , where denotes the odd girth of . Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if then contains as a subgraph a ring graph with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for . Stiebitz et al verified the conjecture for graphs with and . McDonald proved the conjecture when is divisible by . In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every with , and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg‐Seymour conjecture.