Characterizing When Powers of a Caterpillar are Divisor Graphs

Abstract

In this paper, we prove that for any tree $T$, $T^2$ is a divisor graph if and only if $T$ is a caterpillar and the diameter of $T$ is less than six. For any caterpillar $T$ and a positive integer $k\ge 1$ with $diam(T)\le 2k$, we show that $T^k$ is a divisor graph. Moreover, for a caterpillar $T$ and $k\ge 3$ with $diam(T)=2k$ or $diam(T)=2k+1$, we show that $T^k$ is a divisor graph if and only if the centers of $T$ have degree two.