The complexity of frugal colouring

Stefan Bard; Gary MacGillivray; Shayla Redlin

doi:10.1007/s40065-021-00311-7

The complexity of frugal colouring

Arab J Math. 2021;10(1):51-57. doi: 10.1007/s40065-021-00311-7. Epub 2021 Jan 29.

Authors

Stefan Bard¹, Gary MacGillivray¹, Shayla Redlin²

Affiliations

¹ Department of Mathematics and Statistics, University of Victoria, Victoria, BC Canada.
² Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON Canada.

Abstract

A t-frugal colouring of a graph G is an assignment of colours to the vertices of G, such that each colour appears at most t times in the neighbourhood of any vertex. A dichotomy theorem for the complexity of deciding whether a graph has a 1-frugal colouring with k colours was found by McCormick and Thomas, and then later extended to restricted graph classes by Kratochvil and Siggers. We generalize the McCormick and Thomas theorem by proving a dichotomy theorem for the complexity of deciding whether a graph has a t-frugal colouring with k colours, for all pairs of positive integers t and k. We also generalize bounds of Lih et al. for the number of colours needed in a 1-frugal colouring of a given $K_{4}$ -minor-free graph with maximum degree $Δ$ to t-frugal colourings, for any positive integer t.

Keywords: 05C15; 68Q17.