Search for: All records

ABSTRACT A subgraph of a graph with maximum degree is ‐overfullif . Clearly, if contains a ‐overfull subgraph, then its chromatic index is . However, the converse is not true, as demonstrated by the Petersen graph. Nevertheless, three families of graphs are conjectured to satisfy the converse statement: (1) graphs with (the Overfull Conjecture of Chetwynd and Hilton), (2) planar graphs (Seymour's Exact Conjecture), and (3) graphs whose subgraph induced on the set of maximum degree vertices is the union of vertex‐disjoint cycles (the Core Conjecture of Hilton and Zhao). Over the past decades, these conjectures have been central to the study of edge coloring in simple graphs. Progress had been slow until recently, when the Core Conjecture was confirmed by the authors in 2024. This breakthrough was achieved by extending Vizing's classical fan technique to two larger families of trees: the pseudo‐multifan and the lollipop. This paper investigates the properties of these two structures, forming part of the theoretical foundation used to prove the Core Conjecture. We anticipate that these developments will provide insights into verifying the Overfull Conjecture for graphs where the subgraph induced by maximum‐degree vertices has relatively small maximum degree. 

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 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.