467 research outputs found

    Star 5-edge-colorings of subcubic multigraphs

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    The star chromatic index of a multigraph GG, denoted χs(G)\chi'_{s}(G), is the minimum number of colors needed to properly color the edges of GG such that no path or cycle of length four is bi-colored. A multigraph GG is star kk-edge-colorable if χs(G)k\chi'_{s}(G)\le k. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star 77-edge-colorable, and conjectured that every subcubic multigraph should be star 66-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than 7/37/3 is star list-55-edge-colorable. It is known that a graph with maximum average degree 14/514/5 is not necessarily star 55-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than 12/512/5 is star 55-edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap with arXiv:1701.0410

    Goldberg's Conjecture is true for random multigraphs

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    In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph GG, the chromatic index χ(G)\chi'(G) satisfies χ(G)max{Δ(G)+1,ρ(G)}\chi'(G)\leq \max \{\Delta(G)+1, \lceil\rho(G)\rceil\}, where ρ(G)=max{e(G[S])S/2SV}\rho(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)M(n,m) be the probability space consisting of all loopless multigraphs with nn vertices and mm edges, in which mm pairs from [n][n] are chosen independently at random with repetitions. Our result states that, for a given m:=m(n)m:=m(n), MM(n,m)M\sim M(n,m) typically satisfies χ(G)=max{Δ(G),ρ(G)}\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}. In particular, we show that if nn is even and m:=m(n)m:=m(n), then χ(M)=Δ(M)\chi'(M)=\Delta(M) for a typical MM(n,m)M\sim M(n,m). Furthermore, for a fixed ε>0\varepsilon>0, if nn is odd, then a typical MM(n,m)M\sim M(n,m) has χ(M)=Δ(M)\chi'(M)=\Delta(M) for m(1ε)n3lognm\leq (1-\varepsilon)n^3\log n, and χ(M)=ρ(M)\chi'(M)=\lceil\rho(M)\rceil for m(1+ε)n3lognm\geq (1+\varepsilon)n^3\log n.Comment: 26 page

    Minimum Sum Edge Colorings of Multicycles

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    In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems
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