Frugal coloring of graphs revisited

TL;DR

This work revisits frugal coloring by systematically studying the dual notions of ${\alpha}_t^f(G)$ and ${\tt \chi}_t^f(G)$. It establishes fundamental complexity results (${\sf NP}$-hardness for ${\alpha}_t^f$ on bipartite graphs, linear-time solvability on trees), and proves sharp combinatorial bounds linking these invariants to graph size and degree, with new insights for triangle-free and subcubic graphs. A comprehensive treatment of graph products, block graphs, and Nordhaus–Gaddum inequalities yields exact values in several classes (e.g., block graphs, torus grids) and tight bounds for many products, reinforcing the intricate interplay between local frugality and global colorability. The paper also charts several open problems and directions, including characterizations of graphs attaining the basic lower bound and the behavior of frugal coloring on hypercubes, highlighting both theoretical depth and potential algorithmic applications.

Abstract

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 $G$.

Figures (6)

• Figure 1: For each $j\in[3]$, $\chi_{2}^{f}(G_{j})=\chi_{2}^{f}(\overline{G_{j}})=3$.
• Figure 2: When $n=3$ and $m\equiv1$ (mod $3$), the torus graph $T_{m,3}$ has no $2$-frugal coloring with three colors.
• Figure 3: An optimal $2$-frugal coloring of $T_{3,5}$.
• Figure 4: Patterns used for exhibiting an optimal $2$-frugal coloring of $T_{m,5}$ for $m\geq4$.
• Figure 5: $\chi_{2}^{f}$-colorings of the Cartesian and King's grids

Theorems & Definitions (40)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• proof