Abstract. Let [Formula: see text] be a multigraph. The cover index [Formula: see text] of [Formula: see text] is the greatest integer [Formula: see text] for which there is a coloring of [Formula: see text] with [Formula: see text] colors such that each vertex of [Formula: see text] is incident with at least one edge of each color. Let [Formula: see text] be the minimum degree of [Formula: see text], and let [Formula: see text] be the codensity of [Formula: see text], defined by [Formula: see text], where [Formula: see text] is the set of all edges of [Formula: see text] with at least one end in [Formula: see text]. It is easy to see that [Formula: see text]. In 1978, Gupta proposed the following codensity conjecture: Every multigraph [Formula: see text] satisfies [Formula: see text], which is the dual version of the Goldberg–Seymour conjecture on edge-colorings of multigraphs. In this note, we prove that [Formula: see text] if [Formula: see text] is not integral and [Formula: see text] otherwise. We also show that this codensity conjecture implies another conjecture concerning the cover index made by Gupta in 1967.

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $\chi'(G) = \Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $\Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $\Delta(G) > |V(G)|/3$. In this paper, we show that any $\Delta$-critical graph $G$ is overfull if $\Delta(G) - 7\delta(G)/4\ge (3|V(G)|-17)/4$.