Partial Steiner Triple Systems With an Almost Parallel Class Missing Any Given Point

ABSTRACT

We say that a hypergraph is factor critical if it has no 1‐factor but deletion of any vertex results in a hypergraph that has a 1‐factor. We show that any factor‐critical hypergraph of order has at least edges, and that for all integers and with there exists a factor‐critical ‐uniform hypergraph of order that has exactly edges. A partial Steiner triple system is equivalent to a linear 3‐uniform hypergraph. A partial Steiner triple system with the property that for each point there is an almost parallel class missing is equivalent to a factor‐critical linear 3‐uniform hypergraph; we call such a partial Steiner triple system factor critical . For with , we construct a factor‐critical partial Steiner triple system of order with  triples for each integer in the range , where denotes the number of triples in a maximum partial Steiner triple system of order . We also construct factor‐critical partial Steiner triple systems of order with . Since any factor‐critical partial Steiner triple system is nonsequenceable, we consequently obtain some new results on the existence of nonsequenceable partial Steiner triple systems.