Completely independent spanning trees in kth power of 2-connected graphs

Abstract

Let $T_1,T_2,\dots ,T_k$ be spanning trees of a graph $G$. For any two vertices $u,v$ of $G$, if the paths from $u$ to $v$ in these $k$ trees are pairwise openly disjoint, then we say that $T_1,T_2,\dots ,T_k$ are completely independent. Let $G\ncong C_n$, where $n=qk+2,k\ge 6,q\ge 3$. In this paper, we prove that the $k$th power of any 2-connected graph $G$ on $n\ge 2k(k\ge 4)$ vertices has $k$ completely independent spanning trees, unless $G\cong C_n$ where $n=kq+r,r\in \{3,\dots ,k-1\}$ or $n=2k+2,k\ge 6$. The work of this paper improves the results of Hong (2018).