Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs

Abstract

The first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of K_{1,2}-components in a {K_{1,1}, K_{1,2}, C_n:nge 3}-factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are not contained in any {K_{1,1}, K_{1,2}, C_n:nge 3}-factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a {K_{1,1}, K_{1,2}, C_n:nge 3}-factor, and the number of K_{1,2}-components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge e of G, there is a {K_{1,1}, K_{1,2}, C_n:nge 3}-factor F with e in E(F). Consequences of these results for Vizing’s critical graph conjectures are discussed.