Chromatic index determined by fractional chromatic index

TL;DR

This paper proves new bounds under which the chromatic index of a graph equals the ceiling of its fractional chromatic index, advancing understanding of Goldberg's conjecture and related graph coloring problems.

Contribution

It establishes that if the chromatic index exceeds the maximum degree by more than the cube root of half the maximum degree, then it equals the ceiling of the fractional chromatic index, extending previous bounds.

Findings

Proves $oxed{ ext{if } ext{chi'} > ext{Delta} + oot[3]{ ext{Delta}/2} ext{ then } ext{chi'} = oxed{ ext{ceil}( ext{chi}_f')} }

Verifies Goldberg's conjecture for graphs with maximum degree or number of vertices up to 23.

Extends the range of $m$ for which Jakobsen's conjecture holds to $m ext{ } extless= 23$.

Abstract

Given a graph $G$ possibly with multiple edges but no loops, denote by $\Delta$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, $\chi'$ the {\it chromatic index} and $\chi_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $\Delta\le \chi_f' \le \chi' \le \Delta + \mu$, where the upper bound is a classic result of Vizing. While deciding the exact value of $\chi'$ is a classic NP-complete problem, the computing of $\chi_f'$ is in polynomial time. In fact, it is shown that if $\chi_f' > \Delta$ then $\chi_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that $\chi'=\lceil\chi_f'\rceil$ if $\chi'\ge \Delta+2$, which is commonly referred as Goldberg's conjecture. In this paper, we show that if $\chi' >\Delta+\sqrt[3]{\Delta/2}$ then $\chi'=\lceil\chi_f'\rceil$. The previous best known result is for graphs with $\chi'> \Delta +\sqrt{\Delta/2}$ obtained by Scheide, and by Chen, Yu and Zang, independently. It has been shown that Goldberg's conjecture is equivalent to the following conjecture of Jakobsen: {\it For any positive integer $m$ with $m\ge 3$, every graph $G$ with $\chi'>\frac{m}{m-1}\Delta+\frac{m-3}{m-1}$ satisfies $\chi'=\lceil\chi_f'\rceil$.} Jakobsen's conjecture has been verified for $m$ up to 15 by various researchers in the last four decades. We show that it is true for $m\le 23$. Moreover, we show that Goldberg's conjecture holds for graphs $G$ with $\Delta\leq 23$ or $|V(G)|\leq 23$.