Short Brooms in Edge-chromatic Critical Graphs

Abstract

This paper studies short brooms in edge-chromatic critical graphs. We prove that for any short broom in a $\Delta$-critical graph, at most one color is missing at more than one vertex. Moreover, this color (if exists) is missing at exactly two vertices. Applying this result, we verify the Vertex-splitting Conjecture for graphs with $\Delta \geq 2(n-1)/3$ and the Overfull Conjecture for $\Delta$-critical graphs satisfying $\Delta \geq (2n+5\delta-12)/3$.