Goldberg's conjecture is true for random multigraphs

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.