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- W2963917233 abstract "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 χ f ′ the fractional chromatic index of G , respectively. It is known that Δ ≤ χ f ′ ≤ 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 χ f ′ is in polynomial time. In fact, it is shown that if χ f ′ > Δ then χ f ′ = 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 ′ = ⌈ χ f ′ ⌉ 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 ′ > m m − 1 Δ + m − 3 m − 1 satisfies χ ′ = ⌈ χ f ′ ⌉ . 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 / 2 3 then χ ′ = ⌈ χ f ′ ⌉ . 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 ." @default.
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- W2963917233 date "2018-07-01" @default.
- W2963917233 modified "2023-10-15" @default.
- W2963917233 title "Chromatic index determined by fractional chromatic index" @default.
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- W2963917233 doi "https://doi.org/10.1016/j.jctb.2018.01.006" @default.
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