AI 中文总结
本文针对树中极大解离集的最大数量问题，研究Wang等人2025年提出的猜想，给出阶数$n\boldsymbol{\text{≥}}3$的树中该数量的分段函数表达式，并刻画对应的极值树。
AI 中文摘要
设$G$为简单图，Yannakakis于1981年提出的$G$的解离集，定义为诱导子图中每个顶点度数至多为1的顶点集合；若解离集不被任何其他解离集作为真子集包含，则称为极大解离集。2025年，Wang等人[ZiyuanWang]证明：对任意阶数$n\boldsymbol{\text{≥}}4$的树$T$，$T$中极大解离集的数量至多为$3^{\frac{n-1}{3}}+\frac{n-1}{3}$，并刻画了达到该上界的极值树；他们还提出了关于极大解离集上界的猜想。本文研究该猜想，证明阶数$n(\boldsymbol{\text{≥}}3)$的树中极大解离集的最大数量为$g(n)$，其中$g(n)$分段定义为：当$n=3,4,5,6$时，$g(n)=n$；当$n\boldsymbol{\text{≡}}1\boldsymbol{\text{ (mod }}3),n\boldsymbol{\text{≥}}7$时，$g(n)=3^{\frac{n-1}{3}}+\frac{n-1}{3}$；当$n\boldsymbol{\text{≡}}2\boldsymbol{\text{ (mod }}3),n\boldsymbol{\text{≥}}8$时，$g(n)=4·3^{\frac{n-5}{3}}+n-5$；当$n\boldsymbol{\text{≡}}0\boldsymbol{\text{ (mod }}3),n\boldsymbol{\text{≥}}12$且$n≠21$时，$g(n)=16·3^{\frac{n-9}{3}}+3n-25$；当$n=9$时，$g(n)=19$；当$n=21$时，$g(n)=1349$。本文还刻画了具有该最大数量极大解离集的极值树。
英文摘要
Let $G$ be a simple graph. A dissociation set of $G$ proposed by Yannakakis in $1981$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most $1$. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. In $2025$, Wang et al.\cite{ZiyuanWang} established that for any tree $T$ of order $n\geq 4$, the number of maximal dissociation sets in $T$ is at most $3^{\frac{n-1}{3}}+\frac{n-1}{3}$ and characterized the extremal trees attaining the upper bound. They also proposed a conjecture about the upper bound of the maximal dissociation set. In this paper, we consider this conjecture and show that the maximum number of maximal dissociation sets in a tree of order $n(n\geq 3)$ is $g(n)$, where \[ g(n) = \begin{cases} n, & n=3,4,5,6,\\ 3^{\frac{n-1}{3}}+\frac{n-1}{3}, & n \equiv 1 \pmod{3},~n\geq7,\\ 4\cdot 3^{\frac{n-5}{3}}+n-5, & n \equiv 2 \pmod{3},~n\geq8, \\ 16\cdot 3^{\frac{n-9}{3}}+3n-25, & n \equiv 0 \pmod{3},~n\geq12~\text{and }~n\neq21, \\ 19, & n=9, \\ 1349, & n=21. \end{cases} \] We also characterize the extremal trees with the maximum number of maximal dissociation sets.
