Existence and embeddings of partial Steiner triple systems of order ten with cubic leaves

Abstract

Denote the set of 21 non-isomorphic cubic graphs of order 10 by L . We first determine precisely which L∈ L occur as the leave of a partial Steiner triple system, thus settling the existence problem for partial Steiner triple systems of order 10 with cubic leaves. Then we settle the embedding problem for partial Steiner triple systems with leaves L∈ L . This second result is obtained as a corollary of a more general result which gives, for each integer v⩾10 and each L∈ L , necessary and sufficient conditions for the existence of a partial Steiner triple system of order v with leave consisting of the complement of L and v−10 isolated vertices.