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. © 1992.
Cai, L., & Ellis, J. A. (1992). Edge colouring line graphs of unicyclic graphs. Discrete Applied Mathematics, 36(1), 75–82. https://doi.org/10.1016/0166-218X(92)90206-P