Decompositions of $K_v$ Into Four Kinds of Graphs with Eight Vertices and Eight Edges

Abstract

Let $K_v$ be the complete graph with $v$ vertices. Let $G$ be a finite simple graph. A $G$-decomposition of $K_v$, denoted by $(v,G,1)$-GD, is a pair $(X,\mathcal{B})$, where $X$ is the vertex set of $K_v$, and $\mathcal{B}$ is a collection of subgraphs of $K_v$, called blocks, such that each block is isomorphic to $G$. In this paper, the discussed graphs are $G_i$, $i=1,2,3,4$, where $G_i$ are four kinds of graphs with eight vertices and eight edges. We obtain the existence spectrum of $(v,G_i,1)$-GD.