arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1409.3159math.CO
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关于连通图中的大$k$ -端树

On large $k$-ended trees in connected graphs

Zh. G. Nikoghosyan

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15 pages, major revision-2
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摘要：

度为1的顶点称为端点，树的端点称为叶。最多有$k$叶的树称为$k$结束树。对于正整数$k$，令$t_k$是最大$k$结束树的阶。设$σ_m$是$m$顶点的独立集合的最小度和。主要结果（定理2）给出了$t_{k+1}$在$σ_m$和相对阶方面的一个下界：如果$G$是连通图且$k$，$λ$，$m$是正整数且$2\le m\le\min\{k,λ\}+1$，则要么是$t_{k+1} \ge σ_m +λ(k-m+1)+1$，要么是$t_k\ge t_{k+1}-λ+1$ .

英文摘要：

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let $σ_m$ be the minimum degree sum of an independent set of $m$ vertices. The main result (Theorem 2) provides a lower bound for $t_{k+1}$ in terms of $σ_m$ and relative orders: if $G$ is a connected graph and $k$, $λ$, $m$ are positive integers with $2\le m\le\min\{k,λ\}+1$ then either $t_{k+1} \ge σ_m +λ(k-m+1)+1$ or $t_k\ge t_{k+1}-λ+1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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