Least Common Multiple of Path, Star with Cartesian Product of Some Graphs

Abstract:

      A graph $G$ without isolated vertices is a least common multiple of two graphs $H_1$ and $H_2$ if $G$ is a smallest graph, in terms of number of edges, such that there exists a decomposition of $G$ into edge disjoint copies of $H_1$ and $H_2$. The collection of all least common multiples of $ H_1 $ and $ H_2 $ is denoted by $ \LCM (H_1, H_2) $ and the size of a least common multiple of $ H_1 $ and $ H_2 $ is denoted by $ \lcm (H_1, H_2) $. In this paper $\lcm ( P_4, P_m\ \square\ P_n) $, $\lcm (P_4, C_m \ \square\ C_n)$ and $\lcm (K_{1,3}, K_{1,m}\ \square\ K_{1,n}) $ are determined.