Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly - Laurent Beaudou; Caroline Brosse; Oscar Defrain; Florent Foucaud; Aurélie Lagoutte; Vincent Limouzy; Lucas Pastor (Discrete Mathematics & Theoretical Computer Science, 2024)
The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $σ= v_1,…,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $σ$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $χ(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $χ(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.