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- W2963637355 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 ρ(G)=max{e(G[S])⌊|S|/2⌋|S⊆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{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)n3logn, and χ′(M)=⌈ρ(M)⌉ for m≥(1+ε)n3logn. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree." @default.
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- W2963637355 date "2019-09-01" @default.
- W2963637355 modified "2023-09-28" @default.
- W2963637355 title "Goldberg's conjecture is true for random multigraphs" @default.
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- W2963637355 doi "https://doi.org/10.1016/j.jctb.2019.02.005" @default.
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