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arXiv · 2110.14003

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
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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.

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Laurent Beaudou, Caroline Brosse, Oscar Defrain, Florent Foucaud, Aurélie Lagoutte, Vincent Limouzy, Lucas Pastor. 2021-10-26. Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly. https://doi.org/10.46298/dmtcs.8715

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