Chromatic index determined by fractional chromatic index

Abstract

Given a graph G possibly with multiple edges but no loops, denote by ∆ the maximum degree, µ the multiplicity, χ ′ the chromatic index and χ ′ f the fractional chromatic index of G, respectively.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, where the maximality is taken over all induced subgraphs H of G. Gupta (1967), Goldberg (1973), Andersen (1977), andSeymour (1979) , 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 ∆ + 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 ′ ≥ ∆ + 3 ∆/2 then χ ′ = ⌈χ ′ f ⌉. The previous best known result is for graphs with χ ′ > ∆+ ∆/2 obtained by Scheide, and by Chen, Yu and Zang, independently. Moreover, we show that Goldberg's conjecture holds for graphs G with ∆ ≤ 23 or |V (G)| ≤ 23. same color. The chromatic index χ ′ := χ ′ (G) is the least integer k such that G has a k-edge-coloring. In graph edge-coloring, the central question is to determine the chromatic index χ ′ for graphs. We refer the book [17] of Stiebitz, Scheide, Toft and Favrholdt and the elegant survey [12] of McDonald for literature on the recent progress of graph edge-colorings. Clearly, χ ′ ≥ ∆. Conversely, Vizing showed that χ ′ ≤ ∆ + µ, where µ := µ(G) is the multiplicity of G. However, determining the exact value of χ ′ is a very difficult problem. Holyer [8] showed that the problem is NP-hard even restricted to simple cubic graphs. To estimate χ ′ , the notion of fractional chromatic index is introduced.

Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

“…As the statement holds trivially when n = 1, we proceed to the induction step and assume that the statement has been established for n − 1; that is, (3)…”

“…Nevertheless, this method suffers some theoretical limitation when applied to prove the conjecture; the reader is referred to Asplund and McDonald [2] for detailed information. Despite various attempts to extend the Tashkinov trees (see, for instance, [3,4,5,31,36]), the difficulty encountered by the method remains unresolved. Even worse, new problem emerges: it becomes very difficult to preserve the structure of an extended Tashkinov tree under Kempe changes (the most useful tool in edge-coloring theory).…”

Orientation-based edge-colorings and linear arboricity of multigraphs

“…Stiebitz, Scheide, Toft, and Favrholdt [37] have proven a fractional version of Conjecture 1. Much better approximations have been obtained for the case f = 1 of the chromatic index using the tool Tashkinov trees (see [9,10,33,38]). A proof of the case f = 1 has even recently been announced by Chen, Jing, and Zang [8], although it awaits verification.…”

Goldberg's conjecture is true for random multigraphs

“…In [6], and independently [26], it was shown that χ ′ (G) ≤ max{∆+ ∆ 2 , ⌈ρ(G)⌉}. A very recent breakthrough by Chen, Gao, Kim, Postle, and Shan [5] proves the best known upper bound of χ ′ (G) ≤ max{∆ + 3 ∆ 2 , ⌈ρ(G)⌉}. For a more thorough history of this important problem see [30].…”