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

Abstract The first part of the paper studies star-cycle factors graphs. It characterizes a graph G and proves upper bounds for minimum number $$K_{1,2}$$ K1,2 -components in $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">{K1,1,K1,2,Cn:n≥3} -factor . Furthermore, it shows where these components are located with respect to Gallai–Edmonds decomposition edges which not contained any second that every edge-chromatic critical has -factor, is bounded terms its fractional matching number. edge e , there F $$e \in E(F)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">e∈E(F) Consequences results Vizing’s conjectures discussed.