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Connected Greedy Colourings

  • Fabrício Benevides
  • Victor Campos
  • Mitre Dourado
  • Simon Griffiths
  • Robert Morris
  • Leonardo Sampaio
  • Ana Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

A 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.

Keywords

Vertex colouring Greedy colouring Connected greedy colouring 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Fabrício Benevides
    • 1
  • Victor Campos
    • 1
  • Mitre Dourado
    • 2
  • Simon Griffiths
    • 3
  • Robert Morris
    • 3
  • Leonardo Sampaio
    • 4
  • Ana Silva
    • 1
  1. 1.Universidade Federal do Ceará (UFC)FortalezaBrazil
  2. 2.Universidade Federal do Rio de Janeiro (UFRJ)Rio de JaneiroBrazil
  3. 3.Instituto Nacional de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  4. 4.Universidade Estadual do Ceará (UECE)FortalezaBrazil

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