On the 1.1 edge-coloring of multigraphs
SIAM Journal on Discrete Mathematics
Asymptotics of the chromatic index for multigraphs
Journal of Combinatorial Theory Series B
An asymptotic approximation scheme for multigraph edge coloring
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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For the chromatic index @g^'(G) of a (multi)graph G, there are two trivial lower bounds, namely the maximum degree @D(G) and the density W(G)=max"H"@?"G","|"V"("H")"|"="2@?|E(H)|/@?|V(H)|/2@?@?. A famous 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 @g^'(G)==@D(G)+2. The considered class of graphs J can be subdivided into an ascending sequence of classes (J"m)"m"="3, and for m=0 and every graph G satisfying @D(G)=12@e^2 the estimate @g^'(G)=