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

Vizing and Gupta showed that the chromatic index ${\chi }^{\mathrm{\prime }}(G)$ of a graph $G$ is bounded above by $\mathrm{\Delta }(G)+\mu (G)$, where $\mathrm{\Delta }(G)$ and $\mu (G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that ${\chi }^{\mathrm{\prime }}(G)\le \mathrm{\Delta }(G)+\lceil \mu (G)\mathrm{/}\lfloor g(G)\mathrm{/}2\rfloor \rceil$, 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 $g(G)\ge 5$ and ${\chi }^{\mathrm{\prime }}(G)\ge \mathrm{\Delta }+2$, then $G$ must be a ring graph of odd girth.