Chromatic index determined by fractional chromatic index

Given a graph G possibly with multiple edges but no loops, denote by Δ the maximum degree, μ the multiplicity, χ′ the chromatic index and χf′ the fractional chromatic index of G, respectively. It is known that Δ≤χf′≤χ′≤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 χf′ is in polynomial time. In fact, it is shown that if χf′>Δ then χf′=max⁡|E(H)|⌊|V(H)|/2⌋, where the maximality is taken over all induced subgraphs H of G. Gupta (1967), Goldberg (1973), Andersen (1977), and Seymour (1979) conjectured that χ′=⌈χf′⌉ if χ′≥Δ+2, which is commonly referred as Goldberg's conjecture. It has been shown that Goldberg's conjecture is equivalent to the following conjecture of Jakobsen: For any positive integer m with m≥3, every graph G with χ′>mm−1Δ+m−3m−1 satisfies χ′=⌈χf′⌉. Jakobsen's conjecture has been verified for m up to 15 by various researchers in the last four decades. We use an extended form of a Tashkinov tree to show that it is true for m≤23. With the same technique, we show that if χ′≥Δ+Δ/23 then χ′=⌈χf′⌉. The previous best known result is for graphs with χ′>Δ+Δ/2 obtained by Scheide, and by Chen, Yu and Zang, independently. Moreover, we showthat Goldberg's conjecture holds for graphs G with Δ≤23 or |V(G)|≤23.