Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

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 Vizing's 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

A brief history of edge-colorings – with personal reminiscences

In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Dénes König to very recent wor

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

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 algorithmically. 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 $P=NP$