On graphs with girth at least five achieving Steffen's edge coloring bound

TL;DR

This work resolves the extremal structure of graphs attaining Steffen's edge-coloring bound when the girth is at least five. By combining cycle-partition decompositions, fan arguments, and Goldberg–Seymour style lower bounds, the authors show that any critical graph with $g(G)\ge 5$, $\mu(G)\ge \left\lfloor g(G)/2\right\rfloor+1$, and $\chi'(G)=\Delta(G)+\left\lceil\dfrac{\mu(G)}{\left\lfloor g(G)/2\right\right\rfloor}\right\rceil$ must contain a ring graph sharing the same chromatic index; in fact, for $g(G)\ge5$ and $\chi'(G)\ge \Delta(G)+2$, such graphs are ring graphs of odd girth. The proof proceeds through a detailed case analysis by girth and cycle-partition depth, with the $g=5$ case treated separately. The results answer open questions of Stiebitz et al. and strongly tie extremal edge-coloring behavior to ring-graph configurations, clarifying when ring graphs are the unique extremal objects for Steffen's bound in this range."

Abstract

Vizing and Gupta showed that the chromatic index $χ'(G)$ of a graph $G$ is bounded above by $Δ(G) + μ(G)$, where $Δ(G)$ and $μ(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that $χ'(G) \leq Δ(G) + \left\lceil μ(G)/\left\lfloor g(G)/2 \right\rfloor \right\rceil$, where $g(G)$ is the girth of the graph $G$. A {\it ring graph} is a graph obtained from a cycle by duplicating some edges. The equality in Steffen's bound is achieved by ring graphs of the form $μC_g$, obtained from an odd cycle $C_g$ by duplicating each edge $μ$ times. We answer two questions posed by Stiebitz et al. regarding the characterization of graphs which achieve Steffen's bound. In particular, we show that if $G$ is a critical graph which achieves Steffen's bound with $g(G)\geq 5$ and $χ'(G)\geq Δ+2$, then $G$ must be a ring graph of odd girth.

Figures (2)

• Figure 1:
• Figure 2: Cycles in Claim \ref{['cla-tFan']}

Theorems & Definitions (15)

• Theorem 1.1: Steffen MR1754344
• Theorem 1.2
• Theorem 2.1: Chen, Hao, Jing, Yu, Zang
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Claim 3.1
• proof
• Claim 3.2