English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is tocolor the edges of $G$ with the minimum number of colors so that no twoadjacent edges have the same color. This problem can be naturally formulated asan integer program, and its linear programming relaxation is called the {\emfractional edge-coloring problem} (FECP). In the literature, the optimal valueof ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractionalchromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let$\Delta(G)$ be the maximum degree of $G$ and let \[\Gamma(G)=\max\Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rmand \hskip 2mm odd} \Big\},\] where $E(U)$ is the set of all edges of $G$ withboth ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ isa lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G),\, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjecturedthat $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over thepast four decades this conjecture, a cornerstone in modern edge-coloring, hasbeen a subject of extensive research, and has stimulated a significant body ofwork. In this paper we present a proof of this conjecture. Our result impliesthat, first, there are only two possible values for $\chi'(G)$, so an analogueto Vizing's theorem on edge-colorings of simple graphs, a fundamental result ingraph theory, holds for multigraphs; second, although it is $NP$-hard ingeneral to determine $\chi'(G)$, we can approximate it within one of its truevalue, 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 afascinating integer rounding property.

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