Decompositions of \(K_{v}\) Into Four Kinds of Graphs with Eight Vertices and Eight Edges

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.