Approximating the chromatic index of multigraphs

It is well known that if G is a multigraph then χ(G) ≥ χ(G) := max{∆(G), Γ(G)}, where χ(G) is the chromatic index of G, χ(G) is the fractional chromatic index of G, ∆(G) is the maximum degree of G, and Γ(G) = max{2|E(G[U ])|/(|U | − 1) : U ⊆ V (G), |U | ≥ 3, |U | is odd}. The conjecture that χ(G) ≤ max{∆(G) + 1, dΓ(G)e} was made independently by Goldberg (1973), Anderson (1977), and Seymour (1979). Using a probabilistic argument Kahn showed that for any c > 0 there exists D > 0 such that χ(G) ≤ χ(G) + cχ(G) when χ(G) > D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ(G) > b(11∆(G) + 8)/10c; and Scheide recently improved this bound to χ(G) > b(15∆(G) + 12)/14c. We prove this conjecture for multigraphs G with χ(G) > b∆(G) + √ ∆(G)/2c, improving the above mentioned results. Our proof yields an algorithm for edge-coloring any multigraph G using at most max{∆(G) + √ ∆(G)/2, dΓ(G)e} colors, which runs in polynomial time provided that ∆(G) is not part of the input. As a consequence, for multigraphs G with χ(G) > ∆(G) + √ ∆(G)/2 the answer to a 1964 problem of Vizing is affirmative. AMS Subject Classification: Primary 05C15, 05C85; Secondary 05C05, 05C70 Department of Mathematics and Statistics, Georgia State University, Atlanta, GA. This author is partially supported by NSF School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332. This author is partially supported by NSA and by NSFC Project 10628102 Department of Mathematics, University of Hong Kong, Hong Kong, China. This author is supported in part by the Research Grants Council of Hong Kong