Goldberg's conjecture is true for random multigraphs

Penny Haxell, Michael Krivelevich, Gal Kronenberg

Research output: Contribution to journalArticlepeer-review

Abstract

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph G, the chromatic index χ(G) satisfies χ(G)≤max⁡{Δ(G)+1,⌈ρ(G)⌉}, where [Formula presented]. We show that their conjecture (in a stronger form) is true for random multigraphs. Let M(n,m) be the probability space consisting of all loopless multigraphs with n vertices and m edges, in which m pairs from [n] are chosen independently at random with repetitions. Our result states that, for a given m:=m(n), M∼M(n,m) typically satisfies χ(G)=max⁡{Δ(G),⌈ρ(G)⌉}. In particular, we show that if n is even and m:=m(n), then χ(M)=Δ(M) for a typical M∼M(n,m). Furthermore, for a fixed ε>0, if n is odd, then a typical M∼M(n,m) has χ(M)=Δ(M) for m≤(1−ε)n3log⁡n, and χ(M)=⌈ρ(M)⌉ for m≥(1+ε)n3log⁡n. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.

Original languageEnglish
Pages (from-to)314-349
Number of pages36
JournalJournal of Combinatorial Theory. Series B
Volume138
DOIs
StatePublished - Sep 2019

Keywords

  • Chromatic index
  • Edge colouring
  • Random graphs
  • Random multigraphs

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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