Embedding partial extended triple systems and totally symmetric quasigroups (Q1377692)

Let \(K^+_n\) be the complete graph on \(n\) vertices with exactly one loop incident with each vertex (a loop is considered an edge). An extended triple is defined to be either a loop, a loop with an edge attached, or a copy of \(K_3\). A (partial) extended triple system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a set of extended triples defined on the vertex set \(V\) which partitions (a subset of) the edges of \(K^+_n\). Partial extended triple systems and extended triple systems of order \(n\) are denoted \(\text{PETS}(n)\) and \(\text{ETS}(n)\), respectively. A \(\text{PETS}(V, B)\) is said to be embedded in an \(\text{ETS}(V', B')\) if \(V\subseteq V'\) and \(B\subseteq B'\). In 1987, Hoffman and Rodger proved that an \(\text{ETS}(n)\) can be embedded in an \(\text{ETS}(v)\) if and only if \(v\geq 2n\), \(v\) is even if \(n\) is even, and \((v, n)\neq (6k+ 5, 12k+ 12)\). The main result of this paper is that a \(\text{PETS}(n)\) can be embedded in can \(\text{EPS}(v)\) for any \(v\geq 4u+ 6\) with \(v\equiv 2\text{ mod }4\).