A better than “best possible” algorithm to edge color multigraphs
Journal of Algorithms
On the chromatic index of multigraphs and a conjecture of seymour (I)
Journal of Combinatorial Theory Series B
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
Improving a family of approximation algorithms to edge color multigraphs
Information Processing Letters
A sublinear bound on the chromatic index of multigraphs
Discrete Mathematics
An asymptotic approximation scheme for multigraph edge coloring
ACM Transactions on Algorithms (TALG)
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It is well known that if G is a multigraph then 驴驴(G)驴驴驴*(G):=max驴{Δ(G),Γ(G)}, where 驴驴(G) is the chromatic index of G, 驴驴*(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max驴{2|E(G[U])|/(|U|驴1):U驴V(G),|U|驴3,驴|U| is odd}. The conjecture that 驴驴(G)驴max驴{Δ(G)+1,驴Γ(G)驴} was made independently by Goldberg (Discret. Anal. 23:3---7, 1973), Anderson (Math. Scand. 40:161---175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423---460, 1979). Using a probabilistic argument Kahn showed that for any c0 there exists D0 such that 驴驴(G)驴驴驴*(G)+c 驴驴*(G) when 驴驴*(G)D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with 驴驴(G)驴(11Δ(G)+8)/10驴; and Scheide recently improved this bound to 驴驴(G)驴(15Δ(G)+12)/14驴. We prove this conjecture for multigraphs G with $\chi'(G)\lfloor\Delta(G)+\sqrt{\Delta(G)/2}\rfloor$ , improving the above mentioned results. As a consequence, for multigraphs G with $\chi'(G)\Delta(G)+\sqrt {\Delta(G)/2}$ the answer to a 1964 problem of Vizing is in the affirmative.