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Star 5-edge-colorings of subcubic multigraphs

Abstract

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

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