Chromatic index determined by fractional chromatic index

Given a graph $G$ possibly with multiple edges but no loops, denote by $Δ$ the {\it maximum degree}, $μ$ the {\it multiplicity}, $χ'$ the {\it chromatic index} and $χ_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $Δ\le χ_f' \le χ' \le Δ+ μ$, where the upper bound is a classic result of Vizing. While deciding the exact value of $χ'$ is a classic NP-complete problem, the computing of $χ_f'$ is in polynomial time. In fact, it is shown that if $χ_f' > Δ$ then $χ_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that $χ'=\lceilχ_f'\rceil$ if $χ'\ge Δ+2$, which is commonly referred as Goldberg's conjecture. In this paper, we show that if $χ' >Δ+\sqrt[3]{Δ/2}$ then $χ'=\lceilχ_f'\rceil$. The previous best known result is for graphs with $χ'> Δ+\sqrt{Δ/2}$ 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 $χ'>\frac{m}{m-1}Δ+\frac{m-3}{m-1}$ satisfies $χ'=\lceilχ_f'\rceil$.} 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 $Δ\leq 23$ or $|V(G)|\leq 23$.

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