A note on connected greedy edge colouring

TL;DR

This paper investigates the properties of connected greedy edge colouring in graphs, establishing complexity results and bounds for specific graph classes, and comparing it to the traditional chromatic index.

Contribution

It introduces the concept of connected greedy edge colouring, proves NP-hardness of certain decision problems, and provides bounds for bipartite and subcubic graphs.

Findings

Determined NP-hardness of deciding if '(G) > '(G)

Proved '(G) = '(G) for bipartite graphs

Established '(G)  4 for subcubic graphs

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 $\chi'(G)$, 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 $\chi_c'(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $\chi_c'(G)>\chi'(G)$. We prove that $\chi'(G)=\chi_c'(G)$ if $G$ is bipartite, and that $\chi_c'(G)\leq 4$ if $G$ is subcubic.