Average degrees of edge-Delta-critical multigraphs

Let $G$ be a loopless multigraph with maximum degree $\Delta(G)$, average degree $\overline{d}(G)$, density $\Gamma(G)$, and chromatic index $\chi'(G)$. A multigraph $G$ is called edge-$\Delta$-critical if $\Delta(G)=\Delta$, $\chi'(G)=\Delta(G)+1$ and $\chi'(H) \le \Delta(G)$ for every proper subgraph $H\subset G$. Vizing conjectured that if $G$ is an edge-$\Delta$-critical simple graph on $n$ vertices, then $\overline{d}(G) \ge \Delta-1+\tfrac{3}{n}$. Motivated by this, we conjecture that every edge-$\Delta$-critical multigraph $G$ satisfies $\overline{d}(G) \ge \tfrac{2\Delta+2}{3}$, which is best possible. We first give a general lower bound in this direction. For any such graph $G$, \[ \overline{d}(G) \ge \begin{cases} \frac{\sqrt{17}-3}{2}(\Delta+1) & \text{if } \Delta \le 112;\\[4pt] \frac{\Delta+\sqrt{2\Delta-1}}{2} & \text{if } \Delta \ge 113. \end{cases} \] This bound can be further improved under an additional condition on the multiplicity $\mu$. In this case, \[ \overline{d}(G)\ge \min\left\{ \frac{2\mu\Delta+2\mu(2\mu-1)}{4\mu-1},\; \frac{\sqrt{17}-3}{2}(\Delta+1) \right\}. \] We also confirm the conjecture for $\Delta \in \{2,3,4,5,6,7,8\}$. As a consequence, Goldberg's conjecture~\cite{Goldberg1984} holds for $\Delta(G)\in\{2,3,4,5\}$, that is, every multigraph $G$ with $\chi'(G)\ge \Delta(G)+1$ satisfies $\Gamma(G)\ge \Delta(G)$.