Decomposition of $\lambda K_v$ into Kites and $4$-Cycles

Abstract

Given a collection of graphs $\mathcal{H}$, an $\mathcal{H}$-decomposition of $\lambda K_v$ is a decomposition of the edges of $\lambda K_v$ into isomorphic copies of graphs in $\mathcal{H}$. A kite is a triangle with a tail consisting of a single edge. In this paper, we investigate the decomposition problem when $\mathcal{H}$ is the set containing a kite and a $4$-cycle, that is, this paper gives a complete solution to the problem of decomposing $\lambda K_v$ into $r$ kites and $s$ $4$-cycles for every admissible values of $v$, $r,\lambda$, and $s$.