The complexity of frugal colouring

Stefan Bard,Gary Macgillivray,Shayla Redlin

doi:10.1007/s40065-021-00311-7

Stefan Bard, Gary Macgillivray + Show 1 more

Open Access

https://doi.org/10.1007/s40065-021-00311-7

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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 Delta to t-frugal colourings, for any positive integer t.

Highlights

Introduction and backgroundA colouring of a graph G is t -frugal if, for every v ∈ V and for every colour c, the colour c is assigned to at most t vertices of N (v)
Frugal colourings were introduced by Hind, Molloy, and Reed in 1997 [16]. Their main result is that every graph with sufficiently large maximum degree has a log8 -frugal ( + 1)-colouring
Molloy and Reed later proved that every graph G with sufficiently large maximum degree has a (50 log / log log )-frugal

Summary

Introduction and background

A colouring of a graph G is t -frugal if, for every v ∈ V and for every colour c, the colour c is assigned to at most t vertices of N (v). Kratochvíl and Siggers consider degree restrictions an√d show that, for each k ≥ 7, the problem of deciding whether a given graph with maximum degree at most 2 k − 1 has a 1-frugal k-colouring is NP-complete. We generalize the theorem of McCormick and Thomas [24] to all pairs of positive integers t and k by proving: Theorem 1.2 If k ≤ 2, or k = 3 and t = 1, the problem of deciding whether a given graph has a t-frugal k-colouring is solvable in polynomial time. If k = 3 and t ≥ 2, or k ≥ 4 and t ≥ 1, the problem of deciding whether a given graph has a t-frugal k-colouring is NP-complete The proof of this theorem appears in Sect. Results on L(i, j)-labellings, which generalize L(1, 1)-labellings, are surveyed in [5,6]

K4-minor-free graphs

Complexity of proper frugal colouring