Edge colouring line graphs of unicyclic graphs

Abstract The chromatic index problem is known to be NP-complete, even for line graphs. In this paper we show that the chromatic index of the line graph of a unicyclic graph is equal to its maximum degree unless it is an odd cycle. The construction used in the proof implies a linear time algorithm for computing an optimal edge colouring of such a line graph. The results are easily extended to line graphs of graphs in which no two cycles have vertices in common.