Journal of Combinatorial Optimization

, Volume 21, Issue 2, pp 219–246 | Cite as

Approximating the chromatic index of multigraphs

Article

Abstract

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.

Keywords

Multigraph Edge coloring Chromatic index Fractional chromatic index 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of MathematicsUniversity of Hong KongHong KongChina

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