# Fractional matchings, component-factors and edge-chromatic critical graphs

## TL;DR

This paper explores star-cycle factors in graphs, characterizes their structure, and proves that edge-chromatic critical graphs always contain specific factors with bounded components, impacting Vizing's conjectures.

## Contribution

It introduces new characterizations of star-cycle factors and proves the existence of such factors in edge-chromatic critical graphs with bounds related to fractional matchings.

## Key findings

- Characterizes star-cycle factors in graphs.
- Proves existence of specific factors in edge-chromatic critical graphs.
- Provides bounds on components based on fractional matchings.

## 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 $\{K_{1,1}, K_{1,2}, C_n\colon n\ge 3\}$-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.