A Note on Vertex List Colouring

  • Authors:

  • P. E. Haxell

  • Affiliations:

  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (e-mail: pehaxell@math.uwaterloo.ca)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2001

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Abstract

Let k be a positive integer and let G be a graph. Suppose a list S(v) of positive integers is assigned to each vertex v, such that(1) ∣S(v)∣ = 2k for each vertex v of G, and(2) for each vertex v, and each c ∈ S(v), the number of neighbours w of v for which c ∈ S(w) is at most k.Then we prove that there exists a proper vertex colouring f of G such that f(v) ∈ S(v) for each v ∈ V(G). This proves a weak version of a conjecture of Reed.