Document Type

Article

Publication Date

1-1-2023

Abstract

Let G = (V, E) be a multigraph. The 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 codensity of G, defined by Φ (G) = min (equation presented) {2|E+(U)||U|+1 : U ⊆ V, | U| ≥ 3 and odd }, 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) ≤ min{δ (G), ⌊ Φ (G)⌋ }. In 1978, Gupta proposed the following codensity conjecture: Every multigraph G satisfies ξ (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 ξ (G) ≥ min{δ (G) - 1, ⌊ Φ (G)⌋ } if Φ (G) is not integral and ξ (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.

Publication Source (Journal or Book title)

SIAM Journal on Discrete Mathematics

First Page

1666

Last Page

1673

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