Colouring a graph frugally

Abstract

We prove that any graph with maximum degree Δ sufficiently large, has a proper vertex colouring using Δ+1 colours such that each colour class appears at most log8 Δ times in the neighbourhood of any vertex. We also show that for β≥1, the minimum number of colours required to colour any such graph so that each vertex appears at most β times in the neighbourhood of any vertex is θ(Δ+Δ1+1/β/β), showing in particular that when β=logΔ/loglogΔ, such a colouring cannot always be achieved with O(Δ) colours. We also provide a polynomial time algorithm to find such a colouring. This has applications to the total chromatic number of a graph.

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The second two authors were supported by NATO Collaborative Research Grant #CRG950235.

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Hind, H., Molloy, M. & Reed, B. Colouring a graph frugally. Combinatorica 17, 469–482 (1997). https://doi.org/10.1007/BF01195001

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Mathematics Subject Classification (1991)

  • 05C15