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dc.contributor.authorHaxell, Penny
dc.contributor.authorKrivelevich, Michael
dc.contributor.authorKronenberg, Gal 18:39:52 (GMT) 18:39:52 (GMT)
dc.descriptionThe final publication is available at Elsevier via © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
dc.description.abstractIn 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.en
dc.description.sponsorshipNatural Sciences and Engineering Research Councilen
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International*
dc.subjectchromatic indexen
dc.subjectedge coloringen
dc.subjectrandom graphsen
dc.subjectrandom multigraphsen
dc.titleGoldberg's conjecture is true for random multigraphsen
dcterms.bibliographicCitationHaxell, Penny, Michael Krivelevich, and Gal Kronenberg. “Goldberg’s Conjecture Is True for Random Multigraphs.” Journal of Combinatorial Theory, Series B 138 (September 1, 2019): 314–49.
uws.contributor.affiliation1Faculty of Mathematicsen
uws.contributor.affiliation2Combinatorics and Optimizationen

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