Frugal Colouring of Graphs

A $k$-frugal colouring of a graph $G$ is a proper colouring of the vertices of $G$ such that no colour appears more than $k$ times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, planar graphs with large girth, and outerplanar graphs, and relate this parameter with several well-studied colourings, such as colouring of the square, cyclic colouring, and $L(p,q)$-labelling. We also study frugal edge-colourings of multigraphs.

💡 Research Summary

The paper investigates a variant of graph colouring called k‑frugal colouring, introduced by Hind, Molloy and Reed in 1997. In a k‑frugal vertex colouring each vertex receives a proper colour (adjacent vertices receive distinct colours) and, additionally, no colour may appear more than k times in the neighbourhood of any vertex. The smallest number of colours needed for such a colouring is denoted χₖ(G); the corresponding list‑colouring version is chₖ(G). An analogous definition for edges, χ′ₖ(G), is also introduced for multigraphs.

The authors first observe that χ₁(G) coincides with the chromatic number of the square of G, χ(G²), establishing a direct link between frugal colourings and the well‑studied problem of colouring graph squares. Building on Wegner’s conjecture on χ(G²) for planar graphs, they formulate a new conjecture (Conjecture 2.2) that predicts sharp upper bounds for χₖ(G) in terms of the maximum degree Δ and the parameter k. The conjectured bounds are shown to be tight by constructing a family of planar graphs Gₘ (Figure 1) that attain them.

Using a structural lemma of Van den Heuvel and Mguinness (Lemma 2.3), the paper derives a general list‑frugal bound for planar graphs with Δ ≥ 12: \