Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Short Brooms in Edge-chromatic Critical Graphs

Yonglei Chen, Yan Cao

TL;DR

This work analyzes short broom structures within $Δ$-critical graphs to gain precise control over color-missing sets in edge colorings. The authors prove a Main Theorem that bounds the total missing-color count across a short broom, forcing at most one color to be missing at multiple vertices and, if present, at exactly two vertices. Leveraging this bound, they verify the Vertex-splitting Conjecture for graphs with $Δ \geq \frac{2(n-1)}{3}$ and the Overfull Conjecture for $Δ$-critical graphs with $Δ \geq \frac{2n+5δ-12}{3}$, respectively. The results extend the applicability of short broom techniques and advance understanding of class-1 versus class-2 behavior in high-degree graphs, connecting vertex-splitting phenomena with overfull subgraphs via refined Kempe-chain and path/fan arguments.

Abstract

This paper studies short brooms in edge-chromatic critical graphs. We prove that for any short broom in a $Δ$-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 $Δ\geq 2(n-1)/3$ and the Overfull Conjecture for $Δ$-critical graphs satisfying $Δ\geq (2n+5δ-12)/3$.

Short Brooms in Edge-chromatic Critical Graphs

TL;DR

This work analyzes short broom structures within -critical graphs to gain precise control over color-missing sets in edge colorings. The authors prove a Main Theorem that bounds the total missing-color count across a short broom, forcing at most one color to be missing at multiple vertices and, if present, at exactly two vertices. Leveraging this bound, they verify the Vertex-splitting Conjecture for graphs with and the Overfull Conjecture for -critical graphs with , respectively. The results extend the applicability of short broom techniques and advance understanding of class-1 versus class-2 behavior in high-degree graphs, connecting vertex-splitting phenomena with overfull subgraphs via refined Kempe-chain and path/fan arguments.

Abstract

This paper studies short brooms in edge-chromatic critical graphs. We prove that for any short broom in a -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 and the Overfull Conjecture for -critical graphs satisfying .

Paper Structure

This paper contains 7 sections, 18 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Definitions and Preliminary Results
  4. Multifan, Kierstead Path and Short Broom
  5. Proof of Theorem \ref{['ne-broom']}
  6. Proof of Theorem \ref{['vertex-splitting thm']}
  7. Proof of Theorem \ref{['overfull thm']}

Key Result

Theorem 1.1

Let $G$ be a $\Delta$-critical graph with an edge $xy\in E(G)$, and let $\varphi \in \mathcal{C}^{\Delta}(G-xy)$. Suppose $B = (x, xy,$$y, yz, z, zv_1,$$v_1,\ldots, zv_p, v_p)$ is a short broom with respect to $xy$ and $\varphi$. Then

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.1: Main Theorem
  • Conjecture 1: Vertex-splitting Conjecture
  • Theorem 1.2
  • Conjecture 2: Overfull Conjecture
  • Theorem 1.3
  • Lemma 2.1: Vizing's Adjacency Lemma (VAL), VizingCritical1965
  • Definition 2.1
  • Lemma 2.2: GECB, Theorem 2.1
  • Definition 2.2
  • ...and 26 more