Graph Designs for 6-Circle with Two Pendant Edges

Abstract

Let $\lambda K_v$ be the complete multigraph of order $v$ and index $\lambda$, where any two distinct vertices $x$ and $y$ are joined exactly by $\lambda$ edges $\{x,y\}$. Let $G$ be a finite simple graph. A $G$-design of $\lambda K_v$, denoted by $(v,G,\lambda )$-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 $\lambda K_v$, 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 $\mathcal{B}$. There are four graphs which are a 6-circle with two pendant edges, denoted by $G_i$, $i=1,2,3,4$. In [9], we have solved the existence problems of $(v,G_i,1)$-GD. In this paper, we obtain the existence spectrum of $(v,G_i,\lambda )$-GD for any $\lambda >1$.