$${\mathrm{TS}}(v,\lambda )$$ with Cyclic 2-Intersecting Gray Codes: $$v\equiv 0$$ or $$4\pmod {12}$$

Abstract

A $${\mathrm{TS}}(v,{\lambda })$$ is a pair $$(V,\mathcal {B})$$ where V contains v points and $$\mathcal {B}$$ contains 3-element subsets of V so that each pair in V appears in exactly $${\lambda }$$ blocks. A 2-block intersection graph (2-BIG) of a $${\mathrm{TS}}(v,{\lambda })$$ is a graph where each vertex is represented by a block from the $${\mathrm{TS}}(v,{\lambda })$$ and each pair of blocks $$B_i,B_j\in \mathcal {B}$$ are joined by an edge if $$|B_i\cap B_j|=2$$. We show that there exists a $${\mathrm{TS}}(v,{\lambda })$$ for $$v\equiv 0$$ or $$4\pmod {12}$$ whose 2-BIG is Hamiltonian for all admissible $$(v,{\lambda })$$. This is equivalent to the existence of a $${\mathrm{TS}}(v,{\lambda })$$ with a cyclic 2-intersecting Gray code.