Further New Properties of Divisor Graphs

Abstract

In this paper, we characterize the graphs $G$ and $H$ for which the Cartesian product $G\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}H$ is a divisor graph. We show that divisor graphs form a proper subclass of perfect graphs. Additionally, we prove that cycle permutation graphs of order at least 8 are divisor graphs if and only if they are perfect. Some results concerning amalgamation operations for obtaining new divisor graphs from old ones are presented. We view block graphs as vertex amalgams.