Graph Designs for Eight Graphs with Six Vertices and Eight Edges (index \(>1\))

Abstract

Let \(K_v\) be a complete graph with \(v\) vertices, and \(G = (V(G), E(G))\) be a finite simple graph. A \(G\)-design \(G-GD_\lambda(v)\) 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 \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). In this paper, the existence of graph designs \(G-GD_\lambda(v)\), \(\lambda > 1\), for eight graphs \(G\) with six vertices and eight edges is completely solved.