Goldberg's conjecture is true for random multigraphs

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.