Large Sets of $P_3$-Decompositions of Complete Multipartite Graphs

Abstract

Let $H$ and $G$ be two graphs, where $G$ is a simple subgraph of $H$. A $G$-decomposition of $H$, denoted by $(H,G)$-GD, is a partition of all the edges of $H$ into subgraphs ($G$-blocks), each of which is isomorphic to $G$. A large set of $(H,G)$-GD, denoted by $(H,G)$-LGD, is a partition of all subgraphs isomorphic to $G$ of $H$ into $(H,G)$-GDs. In this paper, we determine the existence spectrums for $(\lambda K_{m,n},P_3)$-EGD and $(\lambda K_{n,n,n},P_3)$-LGD.