3 research outputs found

    On Edge Coloring of Multigraphs

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    Let Ξ”(G)\Delta(G) and Ο‡β€²(G)\chi'(G) be the maximum degree and chromatic index of a graph GG, respectively. Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every multigraph GG satisfies Ο‡β€²(G)≀max⁑{Ξ”(G)+1,Ξ“(G)}\chi'(G) \le \max\{ \Delta(G) + 1, \Gamma(G) \}, where Ξ“(G)=max⁑HβŠ†G⌈∣E(H)∣⌊12∣V(H)βˆ£βŒ‹βŒ‰\Gamma(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil is the density of GG. In this paper, we present a polynomial-time algorithm for coloring any multigraph with max⁑{Ξ”(G)+1,Ξ“(G)}\max\{ \Delta(G) + 1, \Gamma(G) \} many colors, confirming the conjecture algorithmically. Since Ο‡β€²(G)β‰₯max⁑{Ξ”(G),Ξ“(G)}\chi'(G)\geq \max\{ \Delta(G), \Gamma(G) \}, this algorithm gives a proper edge coloring that uses at most one more color than the optimum. As determining the chromatic index of an arbitrary graph is NPNP-hard, the max⁑{Ξ”(G)+1,Ξ“(G)}\max\{ \Delta(G) + 1, \Gamma(G) \} bound is best possible for efficient proper edge coloring algorithms on general multigraphs, unless P=NPP=NP
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