A note on connected greedy edge colouring

Abstract

Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index ${\chi }^{\prime }(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let ${\chi }_c^{\prime }(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether ${\chi }_c^{\prime }(G)>{\chi }^{\prime }(G)$. We prove that ${\chi }^{\prime }(G)={\chi }_c^{\prime }(G)$ if $G$ is bipartite, and that ${\chi }_c^{\prime }(G)\le 4$ if $G$ is subcubic.

Comments welcome, 12 pages