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

TL;DR

This work studies connected greedy colourings and the good/bad/ugly taxonomy. It introduces the notion of great graphs and a constructive inductive framework to build good connected orderings, yielding efficient colouring for several classes such as block graphs and cacti. The authors prove that no perfect graph is ugly, and extend the result to $K_4$-minor-free and comparability graphs, providing polynomial-time algorithms to obtain good connected orderings in these cases. The findings deepen our understanding of when greedy First-Fit colourings align with the chromatic number and offer practical algorithms for computing optimal connected orderings in important graph families.

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 $σ= v_1,\dots,v_n$, 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.

Figures (5)

• Figure 1: The fish and the gem, two bad graphs discovered in BT94 and HdW89 (bad connected vertex-orderings are $v_1,\ldots,v_n$).
• Figure 2: An ugly planar cubic graph from BT94.
• Figure 3: (a) An ugly planar claw-free graph from clawfree. (b) An ugly planar line graph.
• Figure 4: A comparability graph (left), and a transitive orientation where minimal elements are represented by white disks and maximal elements by white squares (right).
• Figure 5: The situation of Theorem \ref{['thm:comp']}; $(i)$ a colouring obtained with the method of Golumbic $(ii)$ the same colouring restricted to non-maximal elements $(iii)$ the colouring obtained by swapping colours $2$ and $k$.

Theorems & Definitions (15)

• Lemma 1
• proof
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 6: H-free