$P_{3}$-Factors in the Square of a Tree

Abstract

A spanning subgraph H of a graph G is a $P_{3}$-factor of G if every component of H is a path of order three. The square of a graph G, denoted by $G^{2}$, is the graph with vertex set V(G) such that two vertices are adjacent in $G^{2}$ if and only if their distance in G is at most 2. A graph G is subcubic if it has maximum degree at most three. In this paper, we give a sharp necessary condition for the existence of $P_{3}$-factors in the square of a tree, which improves a result of Li and Zhang (Graphs Combin 24:107–111, 2008). In addition, we will also present a sufficient condition for the existence of $P_{3}$-factors in the square of a tree, which has the following interesting application to subcubic trees: if T is a subcubic tree of order 3n such that $|L(T-L(T))|\le 7$, then $T^{2}$ has a $P_{3}$-factor, where L(T) denotes the set of leaves in T. Examples show that the upper bound 7 on $|L(T-L(T))|$ is sharp.