Goldberg's conjecture is true for random multigraphs
Loading...
Date
2019-09
Authors
Haxell, Penny
Krivelevich, Michael
Kronenberg, Gal
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph G, the chromatic index χ′(G) satisfies χ′(G) ≤ max{∆(G)+1,⌈ρ(G)⌉}, where ρ(G) = max\{\frac {e(G[S])}{\lfloor|S|/2\rfloor} \mid S\subseteq V \}$.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{∆(G),⌈ρ(G)⌉}. In particular, we show that if n is even and m := m(n), then χ′(M) = ∆(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) = ∆(M) for m ≤ (1−ε)n3 logn, and χ′ (M ) = ⌈ρ(M )⌉ for m ≥ (1 + ε)n3 log n. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.
Description
The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2019.02.005. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords
chromatic index, edge coloring, random graphs, random multigraphs