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- W2461203978 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 $Deltale chi_f' le chi' le Delta + mu$, 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 $chi_f'= max frac{|E(H)|}{lfloor |V(H)|/2rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta,(1967), Goldberg,(1973), Andersen,(1977), and Seymour,(1979) conjectured that $chi'=lceilchi_f'rceil$ 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 $chi'=lceilchi_f'rceil$. 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 $mge 3$, every graph $G$ with $chi'>frac{m}{m-1}Delta+frac{m-3}{m-1}$ satisfies $chi'=lceilchi_f'rceil$.} 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 $mle 23$. Moreover, we show that Goldberg's conjecture holds for graphs $G$ with $Deltaleq 23$ or $|V(G)|leq 23$." @default.
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- W2461203978 date "2016-06-25" @default.
- W2461203978 modified "2023-09-29" @default.
- W2461203978 title "Chromatic index determined by fractional chromatic index" @default.
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- W2461203978 doi "https://doi.org/10.48550/arxiv.1606.07927" @default.
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