Short Brooms in Edge-chromatic Critical Graphs

TL;DR

This paper investigates properties of short brooms in edge-chromatic critical graphs, establishing constraints on missing colors and verifying key conjectures for certain classes of these graphs.

Contribution

It introduces new structural results about short brooms in critical graphs and confirms the Vertex-splitting and Overfull Conjectures under specific degree conditions.

Findings

At most one color is missing at more than one vertex in a short broom.

If such a color exists, it is missing at exactly two vertices.

The Vertex-splitting Conjecture is verified for graphs with maximum degree ≥ 2(n-1)/3.

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 $\Delta \geq (2n+5\delta-12)/3$.