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arXiv 2209.09565math.CO
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线图中的完全独立生成树

Completely Independent Spanning Trees in Line Graphs

Toru Hasunuma

备注：
20 pages with 5 figures
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摘要：

图$G$中的完全独立生成树是$G$的生成树，使得对于$G$的任何两个不同的顶点，生成树中它们之间的路径是成对边不相交和内部顶点不相交的。本文给出了$L(G)$中完全独立生成树的最大个数的一个严格下界，其中$L(G)$表示图$G$的线图。基于对具有$k$完全独立生成树的图的一个新的刻画，我们还证明了对于任意的$n \geq 4$阶完全图$K_n$，在$L(K_n)$中存在$\lfloor \frac{n+1}{2} \rfloor$完全独立生成树，其中$\lfloor \frac{n+1}{2} \rfloor$的个数是最优的，使得从$L(K_n)$中删除任意顶点得到的图仍然存在$\lfloor \frac{n+1}{2} \rfloor$完全独立生成树（分别为阶数至多为$n \geq 5$的任何诱导路径）。关于连通性和完全独立的生成树的数量，我们还证明了以下内容，其中$δ(G)$表示$G$的最小度。如果$\bullet$不是超边连通或$δ(G) \geq 2k$，则每个$2k$连通线图$L(G)$都有$k$完全独立的生成树. $\ $ $\bullet$如果$G$是正则的，则每个$(4k-2)$ -连通线图$L(G)$都有$k$完全独立的生成树. $\ $ $\bullet$每个具有$δ(G) \geq k+1$的$(k^2+2k-1)$ -连通线图$L(G)$都有$k$完全独立的生成树.

英文摘要：

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2} \rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees still exist in the graph obtained from $L(K_n)$ by deleting any vertex (respectively, any induced path of order at most $\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where $δ(G)$ denotes the minimum degree of $G$. $\ $ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is not super edge-connected or $δ(G) \geq 2k$. $\ $ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is regular. $\ $ $\bullet$ Every $(k^2+2k-1)$-connected line graph $L(G)$ with $δ(G) \geq k+1$ has $k$ completely independent spanning trees.

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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