图的节俭着色重新审视
Frugal coloring of graphs revisited
AI总结：
本文重新审视了图的节俭着色问题，证明了相关问题的NP难性，并给出了树和多种图类的精确界及色数上界。
摘要：
给定一个图G和一个正整数t，独立集S⊆V(G)是t-节俭的，如果每个顶点在S中的邻居数不超过t。图G的t-节俭着色是将其顶点集划分为t-节俭的独立集。图G中t-节俭独立集的最大基数称为α_t^f(G)，而图G的t-节俭着色的最小基数χ_t^f(G)称为图G的t-节俭色数。节俭着色概念于1998年引入，后来仅在少量论文中被研究。在本文中，我们重新审视这一概念。虽然节俭着色的NP难性已知，但我们证明了α_t^f的决策版本在双色图中也是NP难的，并为树提供了一个线性时间算法来确定其值。我们证明了一个关于χ_{t}^{f}(G)的通用精确下界，该下界以α_{t}^{f}(G)和G的大小表示。我们还给出了任何图G的α_2^f的精确上界，对于最小度数δ≥2的图，这简化为α_2^f(G)≤2n/(δ+2)。我们证明对于任何Δ(G)=3的图，3≤χ_2^f(G)≤5成立。对于若干图类，如块图、Cartesian和强积的多条双向无限路径，我们确定了α_2^f的确切值。我们为所有四种标准图积提供了α_2^f的精确界，这些界以不同不变量表示其因子。最后，我们获得了图G和其补图的2-节俭色数之和的Nordhaus-Gaddum类型不等式，从下和从上界由图G的阶数的函数给出。对于上界χ_{2}^{f}(G)+χ_{2}^{f}(\overline{G})≤3n/2，我们确定了极值图G的家族。
英文摘要：
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 $G$.
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