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

Abstract

AbstractThe 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 $$K_{1,2}$$ K 1 , 2 -components in a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 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 $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor, and the number of $$K_{1,2}$$ 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 $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor F with $$e \in E(F)$$ e ∈ E ( F ) . Consequences of these results for Vizing’s critical graph conjectures are discussed.