Characterizations of amorphic association schemes in terms of fusing triples

Let $\mathcal{R}$ be an association scheme with nontrivial relations $A_1,\ldots,A_d$. We call $\mathcal{R}$ amorphic if every possible fusion of its nontrivial relations gives rise to a fusion scheme. We define the fusing-relations $3$-hypergraph of $\mathcal{R}$ to be the $3$-uniform hypergraph on the vertex set $\{A_1,\ldots,A_d\}$ such that $\{ A_i, A_j, A_k \}$ forms an edge if it fuses, i.e., fusing $A_i, A_j, A_k$ gives rise to a fusion scheme of $\mathcal{R}$. A $3$-uniform hypergraph is called a $3$-sunflower if, for the edges, the union is the set of vertices and the intersection consists of $2$ vertices. In this paper, we prove that for $d\geq 5$, $\mathcal{R}$ is amorphic if its fusing-relations $3$-hypergraph contains two $3$-sunflowers. As a corollary, for $d\geq 5$, $\mathcal{R}$ is amorphic if and only if all triples of its nontrivial relations fuse.