Chromatic index determined by fractional chromatic index

Given a graph $G$ possibly with multiple edges but no loops, denote by $\mathrm{\Delta }$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, ${\chi }^{\mathrm{\prime }}$ the {\it chromatic index} and ${\chi }_f^{\mathrm{\prime }}$ 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 ${\chi }^{\mathrm{\prime }}$ is a classic NP-complete problem, the computing of ${\chi }_f^{\mathrm{\prime }}$ 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 ${\chi }^{\mathrm{\prime }}=\lceil {\chi }_f^{\mathrm{\prime }}\rceil$ if ${\chi }^{\mathrm{\prime }}\ge \mathrm{\Delta }+2$, 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 ${\chi }^{\mathrm{\prime }}>\mathrm{\Delta }+\sqrt{\mathrm{\Delta }\mathrm{/}2}$ 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 ${\chi }^{\mathrm{\prime }}>\frac{m}{m-1}\mathrm{\Delta }+\frac{m-3}{m-1}$ satisfies ${\chi }^{\mathrm{\prime }}=\lceil {\chi }_f^{\mathrm{\prime }}\rceil$.} 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 $\mathrm{\Delta }\le 23$ or $\mathrm{\mid }V(G)\mathrm{\mid }\le 23$.