On Edge Coloring of Multigraphs

Let $\Delta(G)$ and $\chi'(G)$ be the maximum degree and chromatic index of a graph $G$, respectively. Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every multigraph $G$ satisfies $\chi'(G) \le \max\{ \Delta(G) + 1, \Gamma(G) \}$, where $\Gamma(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil$ is the density of $G$. In this paper, we present a polynomial-time algorithm for coloring any multigraph with $\max\{ \Delta(G) + 1, \Gamma(G) \}$ many colors, confirming the conjecture. Since $\chi'(G)\geq \max\{ \Delta(G), \Gamma(G) \}$, 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 $NP$-hard, the $\max\{ \Delta(G) + 1, \Gamma(G) \}$ bound is best possible for efficient proper edge coloring algorithms on general multigraphs, unless $P=NP$.