Partial Steiner Triple Systems Research Articles

Embedding Partial Steiner Triple Systems

We prove that a partial Steiner triple system S of ordern can be embedded in a Steiner triple system T of any given admissible order greater than 4n. Furthermore, if G(S), the missing-edge graph of S, has the property that Δ ( G ) < [ 1 2 ( n + 1 ) ] a n d | E ( G ) | ⩽ ( [ 1 2 ( n - 1 ) ] 2 ) , then S can be embedded in a Steiner triple system of order 2n + 1, provided that 2n + 1 is admissible. We also prove that if there is a partial Steiner triple system of order n with v triples then there is an equitable partial Steiner triple system of order n with v triples. This result, interesting in itself, is used in the proof of the above theorems.

A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6 n + 3

A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6 n + 3

Finite embedding theorems for partial Steiner triple systems

Let P be a finite set and (P, ▪1), (P, ▪2),…, (P, ▪k) any collection of mutually disjoint partial Steiner triple systems. Then these partial triple systems can be embedded in finite mutually disjoint triple systems (S, ▪1), (S, ▪2),…, (S, ▪k). This result is then used to prove the following more general result. If (P, ▪1), (P, ▪2),…, (P, ▪k) are any collection of finite partial Steiner triple systems, then these partial triple systems can be embedded in finite triple systems (S, ▪1), (S, ▪j = ▪i ∩ ▪j for all i ≠ j = 1, 2,…, k.

Disjoint finite partial steiner triple systems can be embedded in disjoint finite steiner triple systems

This paper shows that a pair of disjoint finite partial Steiner triple systems can be embedded in a pair of disjoint finite Steiner triple systems.

Finite partial cyclic triple systems can be finitely embedded

By a cyclic triple system is meant a pair (S, T) where S is a finite set and T is a collection of three element subsets of S arranged cyclically and such that each ordered pair of elements of S appears in exactly one cyclic triplet. By a partial cyclic triple system (PCTS) is meant a pair (S, T) where S is a finite set and T is a collection of three element subsets of S arranged cyclically and such that each ordered pair of elements of S appears in at most one cyclic triplet. Algebraically a CTS is a quasigroup satisfying the identities x 2 =x and x (yx)=y [8], so that algebraically a PCTS is a partial quasigroup Q such that a 2 =a for all a in Q and whenever ab is defined in a then so are b(ab) and (ab)a and further b(ab)=a, (ab)a=b. In what follows we will consider CTS's and PCTS's algebraically. If we omit the words cyclic and ordered in the definition of a CTS and a PCTS we obtain the definition of a Steiner triple system (STS) and a partial Steiner triple system (PSTS) respectively.