On Gupta's Co-density Conjecture

Abstract

Let $G=(V,E)$ be a multigraph. The {\em cover index} $ξ(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. Let $δ(G)$ be the minimum degree of $G$ and let $Φ(G)$ be the {\em co-density} of $G$, defined by \[Φ(G)=\min \Big\{\frac{2|E^+(U)|}{|U|+1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\] where $E^+(U)$ is the set of all edges of $G$ with at least one end in $U$. It is easy to see that $ξ(G) \le \min\{δ(G), \lfloor Φ(G) \rfloor\}$. In 1978 Gupta proposed the following co-density conjecture: Every multigraph $G$ satisfies $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$, which is the dual version of the Goldberg-Seymour conjecture on edge-colorings of multigraphs. In this note we prove that $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$ if $Φ(G)$ is not integral and $ξ(G)\ge \min\{δ(G)-2, \, \lfloor Φ(G) \rfloor-1\}$ otherwise. We also show that this co-density conjecture implies another conjecture concerning cover index made by Gupta in 1967.