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

TL;DR

This paper characterizes graphs with girth at least five that achieve Steffen's refined edge coloring bound, showing they are ring graphs of odd girth if they are critical and meet certain coloring criteria.

Contribution

It answers open questions by characterizing graphs attaining Steffen's bound, especially identifying critical graphs as ring graphs of odd girth under specific conditions.

Findings

Graphs with girth ≥ 5 achieving Steffen's bound are ring graphs of odd girth.

Critical graphs attaining the bound with χ' ≥ Δ+2 are ring graphs of odd girth.

The paper provides a complete characterization of such extremal graphs.

Abstract

Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that $\chi'(G) \leq \Delta(G) + \left\lceil \mu(G)/\left\lfloor g(G)/2 \right\rfloor \right\rceil$, 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 $\mu C_g$, obtained from an odd cycle $C_g$ by duplicating each edge $\mu$ 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 $\chi'(G)\geq \Delta+2$, then $G$ must be a ring graph of odd girth.