ON GUPTA'S CODENSITY CONJECTURE.

Let G=(V,E) be a multigraph. The cover index ε(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 codensity of G, defined by Φ (G) =min... where E+(U) is the set of all edges of G with at least one end in U. It is easy to see that ε (G) ≤ min{ δ (G),Φ (G)}. In 1978, Gupta proposed the following codensity conjecture: Every multigraph G satisfies ε (G) ≥ min{ δ (G) 1,Φ (G)}, which is the dual version of the Goldberg--Seymour conjecture on edge-colorings of multigraphs. In this note, we prove that ε (G) ≥ min{ δ (G) 1, ⌊ Φ (G)⌊ } if Φ (G) is not integral and ε (G) ≥ min{ δ (G) 2, ⌊ Φ (G)⌊ 1} otherwise. We also show that this codensity conjecture implies another conjecture concerning the cover index made by Gupta in 1967. [ABSTRACT FROM AUTHOR]