Journal of Combinatorial Optimization

, Volume 21, Issue 2, pp 219–246

Approximating the chromatic index of multigraphs

Article

DOI: 10.1007/s10878-009-9232-y

Cite this article as:
Chen, G., Yu, X. & Zang, W. J Comb Optim (2011) 21: 219. doi:10.1007/s10878-009-9232-y

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)+cχ*(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 

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

Personalised recommendations