Frugal coloring of graphs revisited

Given a graph $G$ and a positive integer $t$, an independent set $S\subseteq V(G)$ is $t$-frugal if every vertex has at most $t$ neighbors in $S$. A $t$-frugal coloring of $G$ is a partition of its vertex set into $t$-frugal independent sets. The maximum cardinality of a $t$-frugal independent set in $G$ is denoted by $α_t^f(G)$, while the minimum cardinality of a $t$-frugal coloring of $G$, $χ_t^f(G)$, is called the $t$-frugal chromatic number of $G$. Frugal colorings were introduced in 1998 and studied later in just a handful of papers. In this paper, we revisit this concept. While the NP-hardness of frugal coloring is known, we prove that the decision version of $α_t^f$ is NP-complete even for bipartite graphs, and present a linear-time algorithm to determine its value for trees. We prove a general sharp lower bound on $χ_{t}^{f}(G)$ expressed in terms of $α_{t}^{f}(G)$ and size of $G$. We also give a sharp upper bound on the $α_2^f$ of any graph $G$, which in the case of graphs with minimum degree $δ\geq2$ simplifies to $α_2^f(G)\le 2n/(δ+2)$. We prove that $3\leχ_2^f(G)\le 5$ holds for any graph $G$ with $Δ(G)=3$. For several classes of graphs such as block graphs, the Cartesian and strong products of multiple two-way infinite paths, we determine the exact values of $α_2^f$. We provide sharp bounds on the $α_2^f$ in all four standard graph products, which are expressed as different invariants of their factors. Finally, we obtain Nordhaus-Gaddum type inequalities for the sum of the $2$-frugal chromatic numbers of $G$ and its complement from below and from above by functions of the order of $G$. For the upper bound $χ_{2}^{f}(G)+χ_{2}^{f}(\overline{G})\leq 3n/2$, we characterize the family of extremal graphs $G$.

💡 Research Summary

This paper revisits the concept of frugal coloring, a variant of vertex coloring introduced by Hind, Molloy, and Reed in 1997. For a graph G and a positive integer t, a set S⊆V(G) is called t‑frugal if every vertex of G has at most t neighbors in S. A t‑frugal coloring partitions V(G) into t‑frugal independent sets; the largest size of such a set is denoted αₜᶠ(G) and the smallest number of parts in a t‑frugal coloring is χₜᶠ(G).

The authors first address computational complexity. While it was already known that determining χₜᶠ(G) is NP‑complete for any fixed t, they prove that the decision problem “does G contain a t‑frugal independent set of size at least k?” is NP‑complete even when G is bipartite. The reduction is from Exact Cover by 3‑Sets (X3C): each 3‑element subset becomes a path of length three, each element becomes a star with t‑1 leaves, and edges are added between element‑centers and the appropriate path endpoints. The construction forces a maximum t‑frugal independent set to correspond exactly to an exact cover, establishing hardness for all t≥1.

In contrast, for trees the authors give a linear‑time algorithm that computes αₜᶠ(T) for any t. The algorithm roots the tree at a non‑leaf vertex, processes vertices in a bottom‑up order (from farthest to the root toward the root), and greedily adds a vertex to the current set whenever doing so does not violate the t‑frugal independence condition. An inductive proof shows that the greedy set is always extendable to a maximum t‑frugal independent set, yielding an O(|V|) algorithm.

Next, the paper derives several structural bounds. They prove a general lower bound on the t‑frugal chromatic number: \