$(2,C)$-ordered path designs $P(\nu ,3,1)$

Abstract

Let $C$ be the underlying graph of a configuration of $l$ blocks in a path design of order $v$ and block size $3$, $(V,\mathcal{B})$. We say that $(V,\mathcal{B})$ is $(l,C)$-ordered if it is possible to order its blocks in such a way that each set of $l$ consecutive blocks has the same underlying graph $C$. In this paper, we completely solve the problem of the existence of a $(2,C)$-ordered path design $P(v,3,1)$ for any configuration having two blocks.