Star chromatic index
arXiv:1011.3376
Abstract
The star chromatic index $Ï_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree $Î=Î(G)$. Our best lower bound on $Ï_s'$ in terms of $Î$ is $2Î(1+o(1))$ valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.
16 pages, 3 figures