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.