, Volume 21, Issue 2, pp 219-246
Date: 29 May 2009

Approximating the chromatic index of multigraphs

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It is well known that if G is a multigraph then χ′(G)≥χ*(G):=max {Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ*(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max {2|E(G[U])|/(|U|−1):UV(G),|U|≥3, |U| is odd}. The conjecture that χ′(G)≤max {Δ(G)+1,⌈Γ(G)⌉} 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 c>0 there exists D>0 such that χ′(G)≤χ*(G)+c χ*(G) when χ*(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>(11Δ(G)+8)/10; and Scheide recently improved this bound to χ′(G)>(15Δ(G)+12)/14. 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.

G. Chen is partially supported by NSF.
X. Yu is partially supported by NSA and by NSFC Project 10628102.
W. Zang is supported in part by the Research Grants Council of Hong Kong.