Complete Simultaneous Metamorphosis of \(\lambda\)-Fold Kite Systems

Giorgio Ragusa1
1Dipartimento di Matematica e Informatica Universita di Catania viale A. Doria, 6 95125 Catania, Italia

Abstract

Let \((X, \mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition of \(\lambda H\). Let \(G_i\), \(i=1,\ldots,\mu\), be all nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i \mid B \in \mathcal{B}\}\), where \(B_i\) is a subgraph of \(B\) isomorphic to \(G_i\). A complete simultaneous metamorphosis of \((X, \mathcal{B})\) is a rearrangement, for each \(i = 1, \ldots, \mu\), of the edges of \(\bigcup_{B \in \mathcal{B}} (E(B) \setminus E(B_i))\) 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 a \(\lambda\)-fold kite system having a complete simultaneous metamorphosis.