Decompositions of \(K_v\) into Graphs with \(7\) Points, \(7\) Edges, and a \(5\)-Cycle

Abstract

Let \(K_v\) be the complete graph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by exactly one edge \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design 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 \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, the discussed graphs are \(G_i\), \(i = 1,2,3,4\), where \(G_i\) are the four graphs with 7 points, 7 edges, and a 5-cycle. We obtain the existence spectrum of \((v, G_i,1)\)-GD.