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- W2791768847 abstract "In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $chi'(G)$ satisfies $chi'(G)leq max {Delta(G)+1, lceilrho(G)rceil}$, where $rho(G)=max {frac {e(G[S])}{lfloor |S|/2rfloor} mid Ssubseteq V }$. 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 $m:=m(n)$, $Msim M(n,m)$ typically satisfies $chi'(G)=max{Delta(G),lceilrho(G)rceil}$. In particular, we show that if $n$ is even and $m:=m(n)$, then $chi'(M)=Delta(M)$ for a typical $Msim M(n,m)$. Furthermore, for a fixed $varepsilon>0$, if $n$ is odd, then a typical $Msim M(n,m)$ has $chi'(M)=Delta(M)$ for $mleq (1-varepsilon)n^3log n$, and $chi'(M)=lceilrho(M)rceil$ for $mgeq (1+varepsilon)n^3log n$." @default.
- W2791768847 created "2018-03-29" @default.
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- W2791768847 date "2018-03-02" @default.
- W2791768847 modified "2023-09-23" @default.
- W2791768847 title "Goldberg's Conjecture is true for random multigraphs" @default.
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- W2791768847 doi "https://doi.org/10.48550/arxiv.1803.00908" @default.
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