Ring graphs and Goldberg's bound on chromatic index

Yan Cao, Guantao Chen, Shushan He, Guangming Jing

Research output: Contribution to journalArticle

Abstract

Let G be a multigraph with maximum degree Δ and chromatic index χ′. If G is bipartite then χ′ = Δ. Otherwise, by a theorem of Goldberg, χ′ ≤ Δ + 1 + ⌊(Δ − 2)/(go − 1)⌋, where go denotes the odd girth of G. Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if χ′ = Δ + 1 + ⌊(Δ − 2)/(go − 1)⌋ then G contains as a subgraph a ring graph R with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for go = 3. Stiebitz et al verified the conjecture for graphs with go = 5 and Δ ≥ 10. McDonald proved the conjecture when Δ − 2 is divisible by go − 1. In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every go ≥ 3 with Δ ≥ (go − 2go + 5)/2, and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg-Seymour conjecture.

Original languageEnglish (US)
Pages (from-to)440-449
Number of pages10
JournalJournal of Graph Theory
Volume93
Issue number3
DOIs
Publication statusPublished - Mar 1 2020

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Keywords

  • chromatic index
  • critical chromatic graph
  • maximum degree
  • odd girth

ASJC Scopus subject areas

  • Geometry and Topology

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