Star chromatic index

Abstract

The star chromatic index ${\chi }_s^{\prime }(G)$ 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 $\Delta =\Delta (G)$. Our best lower bound on ${\chi }_s^{\prime }$ in terms of $\Delta$ is $2\Delta (1+o(1))$ 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