Given a multigraph $G=(V,E)$, the {em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {em chromatic index} (resp. {em fractional chromatic index}) of $G$, denoted by $chi(G)$ (resp. $chi^*(G)$). Let $Delta(G)$ be the maximum degree of $G$ and let [Gamma(G)=max Big{frac{2|E(U)|}{|U|-1}:,, U subseteq V, ,, |U|ge 3 hskip 2mm {rm and hskip 2mm odd} Big},] where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $max{Delta(G), , lceil Gamma(G) rceil }$ is a lower bound for $chi(G)$. As shown by Seymour, $chi^*(G)=max{Delta(G), , Gamma(G)}$. In the 1970s Goldberg and Seymour independently conjectured that $chi(G) le max{Delta(G)+1, , lceil Gamma(G) rceil}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $chi(G)$, so an analogue to Vizings theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $chi(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $Gamma(G)>Delta(G)$; third, every multigraph $G$ satisfies $chi(G)-chi^*(G) le 1$, so FECP has a fascinating integer rounding property.
تحميل البحث