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

Abstract

Let \((X,\mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition and let \(G_i\), \(i = 1,\ldots,\mu\), be nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i | B \in \mathcal{B}\}\), where \(\mathcal{B_i}\) is a subgraph of \(B\)  isomorphic to \(G_i\). A \(\{G_1,G_2,\ldots,G_\mu\}\)-metamorphosis of \((X,\mathcal{B})\) is a rearrangement, for each \(i=1,\ldots,\mu\), 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.