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.