Kite-Designs Intersecting in Pairwise Disjoint Blocks

Abstract

A kite-design of order $n$ is a decomposition of the complete graph $K_n$ into kites. Such systems exist precisely when $n\equiv 0,1(\mathrm{mod}8)$. Two kite systems $(X,{\mathcal{K}}_1)$ and $(X,{\mathcal{K}}_2)$ are said to intersect in $m$ pairwise disjoint blocks if $|{\mathcal{K}}_1\cap {\mathcal{K}}_2|=m$ and all blocks in ${\mathcal{K}}_1\cap {\mathcal{K}}_2$ are pairwise disjoint. In this paper, we determine all the possible values of $m$ such that there are two kite-designs of order $n$ intersecting in $m$ pairwise disjoint blocks, for all $n\equiv 0,1(\mathrm{mod}8)$.