Abstract

• In this paper, we show that the square graph of a tree T has a spanning tree of maximum degree at most three and with at most max{0, ∑ x∈W (T ) (tT (x) − 2) − 2}vertices of degree three, where 3 W3(T ) = {x ∈V (T ) : there are at least three edge-disjoint paths of length at least two that start x} and tT (x) is the number of edge-disjoint paths with length at least two that start at a vertex x.