The complexity of frugal colouring

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 .