Graph edge colouring: Tashkinov trees and Goldbergʹs conjecture

For the chromatic index χ ′ ( G ) of a (multi)graph G, there are two trivial lower bounds, namely the maximum degree Δ ( G ) and the density W ( G ) = max H ⊆ G , | V ( H ) | ⩾ 2 ⌈ | E ( H ) | / ⌊ | V ( H ) | / 2 ⌋ ⌉ . us conjecture due to Goldberg [M.K. Goldberg, On multigraphs of almost maximal chromatic class, Diskret. Analiz 23 (1973) 3–7 (in Russian)] and Seymour [P.D. Seymour, Some unsolved problems on one-factorization of graphs, in: J.A. Bondy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, New York, 1979] says that every graph G satisfies χ ′ ( G ) ⩽ max { Δ ( G ) + 1 , W ( G ) } . This means that χ ′ ( G ) = W ( G ) for every graph G with χ ′ ( G ) ⩾ Δ ( G ) + 2 . The considered class of graphs J can be subdivided into an ascending sequence of classes ( J m ) m ⩾ 3 , and for m ⩽ 13 the conjecture is already proved. The “last” step was done by Favrholdt, Stiebitz and Toft [L.M. Favrholdt, M. Stiebitz, B. Toft, Graph edge colouring: Vizingʹs theorem and Goldbergʹs conjecture, DMF-2006-10-003, IMADA-PP-2006-20, University of Southern Denmark, preprint] in 2006, using and extending results from Tashkinov [V.A. Tashkinov, On an algorithm to colour the edges of a multigraph, Diskret. Analiz 7 (2000) 72–85 (in Russian)]. These methods are based on a colouring structure called Tashkinov tree. In this paper the same methods are used and extended to handle the “next” step m ⩽ 15 . This leads to the result χ ′ ( G ) ⩽ max { ⌊ 15 14 Δ ( G ) + 12 14 ⌋ , W ( G ) } for every graph G. rmore, the used methods also lead to several improvements of other known upper bounds for the chromatic index. In particular, an asymptotic approximation of the chromatic index can be obtained. We prove that for every ϵ > 0 and every graph G satisfying Δ ( G ) ⩾ 1 2 ϵ 2 the estimate χ ′ ( G ) ⩽ max { ( 1 + ϵ ) Δ ( G ) , W ( G ) } holds. This extends a result of Kahn [J. Kahn, Asymptotics of the chromatic index for multigraphs, J. Combin. Theory Ser. B 68 (1996) 233–254] as well as a result of Sanders and Steurer [P. Sanders, D. Steurer, An asymptotic approximation scheme for multigraph edge coloring, in: Proceedings of the Sixteenth ACM-SIAM Symposium on Discrete Algorithms (SODA05), em SIAM, 2005, pp. 897–906].