Decomposing Complete Graphs into Isomorphic Subgraphs with Six Vertices and Seven Edges

Abstract

Let \(K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-design of \(K_v\), denoted by \(G\)-GD(\(v\)), is a pair of (\(X\), \(\mathcal{B}\)), 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 \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, the discussed graphs are sixteen graphs with six vertices and seven edges. We give a unified method for constructing such \(G\)-designs.