## 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)n^{3}logn, and χ^{′}(M)=⌈ρ(M)⌉ for m≥(1+ε)n^{3}logn. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.

Original language | English |
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Pages (from-to) | 314-349 |

Number of pages | 36 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 138 |

DOIs | |

State | Published - 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