Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

TL;DR

This paper investigates connected greedy colourings in graphs, proving that perfect graphs and certain subclasses are always good, meaning they admit optimal connected greedy colourings, and provides polynomial algorithms for these cases.

Contribution

It proves that no perfect graph is ugly and offers constructive, polynomial-time algorithms for finding optimal connected greedy colourings in specific graph classes.

Findings

No perfect graph is ugly.

Polynomial algorithms exist for good connected orderings.

Certain subclasses like block and comparability graphs are also always good.

Abstract

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 $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ 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 $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ 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.