On the Bounds for Sizes of Least Common Multiples of Several Pairs of Graphs

Abstract

G. Chartrand et al. [3] define a graph $G$ without isolated vertices to be the least common multiple (lcm) of two graphs $G_1$ and $G_2$ if $G$ is a graph of minimum size such that $G$ is both $G_1$-decomposable and $G_2$-decomposable. A bi-star $B_{m,n}$ is a caterpillar with spine length one. In this paper, we discuss a good lower bound for $lcm(B_{m,n},G)$, where $G$ is a simple graph. We also investigate $lcm(B_{m,n},r{K}_2)$ and provide a good lower bound and an appropriate upper bound for $lcm(B_{m,n},P_{r+1})$ for all $m\ge 1$, $n\ge 1$, and $r\ge 1$.