Multi-Decomposition of Complete Graphs into Stars and Bowties of Size 6

Let [Formula: see text] denote a complete graph on [Formula: see text] vertices and [Formula: see text] denote a complete bipartite graph [Formula: see text]. A Bowtie [Formula: see text] is a graph formed by the union of two cycles [Formula: see text] and [Formula: see text] intersecting at a common vertex. A decomposition of a graph [Formula: see text] is a collection of edge-disjoint subgraphs [Formula: see text], such that every edge of G belongs to exactly one [Formula: see text]. Given non-isomorphic subgraphs [Formula: see text] and [Formula: see text] of [Formula: see text], a [Formula: see text] — multi-decomposition of [Formula: see text] is the decomposition of [Formula: see text] into [Formula: see text] copies of [Formula: see text] and [Formula: see text] copies of [Formula: see text], such that [Formula: see text], for some integers [Formula: see text]. In this paper, the multi-decomposition of [Formula: see text] into [Formula: see text] and [Formula: see text] has been investigated and obtained a necessary and sufficient condition when [Formula: see text]. It is proved that for a given positive integer [Formula: see text], [Formula: see text] can be decomposed into [Formula: see text] copies of [Formula: see text] and [Formula: see text] copies of [Formula: see text] for some pair of non-negative integers [Formula: see text] if and only if [Formula: see text], for all [Formula: see text].