Connected Greedy Colourings

Abstract

AbstractA connected vertex ordering of a graph G is an ordering v 1 < v 2 < ⋯ < v n of V(G) such that v i has at least one neighbour in {v 1, …, v i − 1}, for every i ∈ {2, …, n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G) = 2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k ) > χ(G k ), for every k ≥ 3. We then prove that for every graph G, χ c (G) ≤ χ(G) + 1. Finally, we prove that deciding if χ c (G) = χ(G), given a graph G, is a NP-hard problem.KeywordsVertex colouringGreedy colouringConnected greedy colouring