(Δ --k)-critical graphs

Authors:
Babak Farzad;Michael Molloy;Bruce Reed
Affiliations:
Department of Computer Science, University of Toronto, Toronto, Canada M5S 1A1;Department of Computer Science, University of Toronto, Toronto, Canada M5S 1A1 and Microsoft Research, Canada;School of Computer Science, McGill University and CNRS, France
Venue:
Journal of Combinatorial Theory Series B
Year:
2005

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Abstract

Every graph G of maximum degree Δ is (Δ + 1)-colourable and a classical theorem of Brooks states that G is not Δ-colourable iff G has a (Δ + 1)-clique or Δ = 2 and G has an odd cycle. Reed extended Brooks' Theorem by showing that if Δ(G) ≥ 1014 then G is not (Δ - 1)-colourable iff G contains a Δ-clique. We extend Reed's characterization of (Δ - 1)-colourable graphs and characterize (Δ - 2), (Δ - 3), (Δ - 4) and (Δ - 5)-colourable graphs, for sufficiently large Δ, and prove a general structure for graphs with χ close to Δ. We give a linear time algorithm to check the (Δ - k)-colourability of a graph, for sufficiently large Δ and any constant k.