$\{C_4,K_3+e\}$-metamorphosis of $S_{\lambda }(2,4,n)$

Abstract

Let $(X,\mathcal{B})$ be a $\lambda$-fold $G$-decomposition and let $G_i$, $i=1,\dots ,\mu$, be nonisomorphic proper subgraphs of $G$ without isolated vertices. Put ${\mathcal{B}}_i=\{B_i|B\in \mathcal{B}\}$, where ${\mathcal{B}}_{\mathcal{i}}$ is a subgraph of $B$  isomorphic to $G_i$. A $\{G_1,G_2,\dots ,G_{\mu }\}$-metamorphosis of $(X,\mathcal{B})$ is a rearrangement, for each $i=1,\dots ,\mu$, of the edges of $\bigcup _{B\in B}(E(B)\setminus {\mathcal{B}}_i))$ into a family ${\mathcal{F}}_i$ of copies of $G_i$ with a leave $L_i$, such that $(X,{\mathcal{B}}_i\cup {\mathcal{F}}_i,L_i)$ is a maximum packing of $\lambda H$ with copies of $G_i$. In this paper, we give a complete answer to the existence problem of an $S_{\lambda }(2,4,7)$ having a $\{C_4,K_3+e\}$-metamorphosis.