Two Completely Independent Spanning Trees of $P_4$-Free Graphs

Abstract

A graph without induced subgraphs isomorphic to a path of length 3 is $P_4$-free. If a graph G contains two spanning trees $T_1,T_2$ such that for each two distinct vertices x, y of G,  the (x, y)-path in each $T_i$ has no common edge and no common vertex except for the two ends, then $T_1,T_2$ are called two completely independent spanning trees (CISTs) of $G, i\in \{1,2\}.$ Several results have shown that some sufficient conditions for Hamiltonian graphs may also guarantee the existence of two CISTs. Jung proved that a $P_4$-free graph with at least 3 vertices is Hamiltonian if and only if it is 1-tough. Inspired by these results, in this paper, we prove that a $P_4$-free graph G contains two CISTs if and only if G is a 2-connected graph of order $n\ge 4$ and $G\notin \mathcal {K},$ where $\mathcal {K}$ is a family of some graphs. Moreover, we obtain that every 1-tough $P_4$-free graph of order $n\ge 4$ with $G\notin \mathcal {K}'$ contains two CISTs, where $\mathcal {K}'$ is a family of four graphs.