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 \(n \equiv 0,1 \pmod{8}\). 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 \(n \equiv 0,1 \pmod{8}\).