Completely Independent Spanning Trees in Some Regular Graphs

TL;DR

This paper investigates the maximum number of completely independent spanning trees in certain regular, highly connected graphs, revealing that the maximum number can be less than the number of trees considered.

Contribution

It determines the maximum number of completely independent spanning trees in specific 2k-regular, 2k-connected graphs, showing it can be less than k.

Findings

Maximum number of such trees is less than k in some graphs.

Results apply to Cartesian products of complete graphs and cycles.

Provides bounds and exact counts for specific graph classes.

Abstract

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices, except the vertices $u$ and $v$, then $T_1,\ldots, T_k$ are completely independent spanning trees in $G$. For $2k$-regular graphs which are $2k$-connected, such as the Cartesian product of a complete graph of order $2k-1$ and a cycle and some Cartesian products of three cycles (for $k=3$), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always $k$.