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 $\{ K_{ 1,1}, K_{ 1,2}, C_n\colon n\ge 3\}$-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 $\{ K_{ 1,1}, K_{ 1,2}, C_n\colon n\ge 3\}$-factor of $G$. The second part of the paper proves that every edge-chromatic critical graph $G$ has a $\{ K_{ 1,1}, K_{ 1,2}, C_n\colon n\ge 3\}$-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 $\{ K_{ 1,1}, K_{ 1,2}, C_n\colon n\ge 3\}$-factor $F$ with $e \in E(F)$. Consequences of these results for Vizing's critical graph conjectures are discussed.
Categories: math.CO cs.DM