Double Vizing fans in critical class two graphs
Yan Cao, Guantao Chen, Xuli Qi · Journal of Graph Theory · 2022
Abstract Let be a simple graph and be the chromatic index of . We call a ‐critical graph if for every edge of , where is maximum degree of . Let be an edge of ‐critical graph and be an (proper) edge ‐coloring of . An e‐fan is 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.