Cyclic bi‐embeddings of Steiner triple systems on 12s + 7 points

A cyclic face 2‐colourable triangulation of the complete graph K_n in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi‐embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi‐embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi‐embedding and we show how all non‐isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001     

Cyclic Biembeddings of Twofold Triple Systems

We construct face two-colourable triangulations of the graph 2K _n in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ≡ 4 (mod 12). The biembeddings have a cyclic automorphism and the construction employs index 1 current graphs.     

Surface embeddings of Steiner triple systems

A Steiner triple system of order n (STS(n)) is said to be embeddable in an orientable surface if there is an orientable embedding of the complete graph K_n whose faces can be properly 2-colored (say, black and white) in such a way that all black faces are triangles and these are precisely the blocks of the STS(n). If, in addition, all white faces are triangular, then the collection of all white triangles forms another STS(n); the pair of such STS(n)s is then said to have an (orientable) bi-embedding. We study several questions related to embeddings and bi-embeddings of STSs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 325–336, 1998     

Bi-Embeddings of Steiner Triple Systems of Order 15

 It was shown by Gerhard Ringel that one of the three non-isomorphic Steiner triple systems of order 15 having an automorphism of order 5 may be bi-embedded as the faces of a face 2-colourable triangular embedding of K ₁₅ in a suitable orientable surface. Ringel's bi-embedding was obtained from an appropriate current graph. In a recent paper, the present authors showed that a second STS(15) of this type may also be bi-embedded. In the present paper we show that this second bi-embedding may also be obtained from a current graph. Furthermore, we exhibit a third current graph which yields a bi-embedding of the third STS(15) of this type. Received: October 5, 1998     

Automorphism free latin square graphs

In this paper, we show that there exists an automorphism free latin square graph of order n for all n ? 7 and that the number of such graphs goes to infinity with n. These results are then applied to the construction of automorphism free Steiner triple systems.     

A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs

We prove a theorem that for an integer s?0, if 12s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K_n, where n=(12s+7)(6s+7), is at least . By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least 2^αn?+o(n?), ?>1, nonisomorphic nonorientable triangular embeddings of K_n for n=6t+1, . To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given.     

Minimum genus embeddings of the complete graph

In this paper,the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied.There are three approaches to solve this problem.The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths;the second approach is to find a current assignment of the current graph by the theory of current graph;the third approach is to find exponentially many embedding(or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph.According to this three approaches,we can construct exponentially many minimum genus embeddings of complete graph K_(12s+8) in orientable surfaces,which show that there are at least 10/3×(200/9)~s distinct minimum genus embeddings for K_(12s+8) in orientable surfaces.We have also proved that K_(12s+8) has at least 10/3×(200/9)~s distinct minimum genus embeddings in non-orientable surfaces.     

A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2-(mn²+1,mn,(n+1)(mn−1)) design starting from a Steiner system S(2,m+1,mn²+1) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.     

Codes and Anticodes in the Grassman Graph

Perfect codes and optimal anticodes in the Grassman graph G_q(n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and “real” Steiner systems.     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Linearly Derived Steiner Triple Systems

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.     

Steiner triple systems satisfying the 4-vertex condition

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled.We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.     

A construction of disjoint steiner triple systems

A set of n ? 2 disjoint Steiner triple systems on n objects is constructed whenever n has the property that the order of 2 modulo n ? 2 is an odd number.     

Orientable imbedding of line graphs

In this paper we compute the orientable genus of the line graph of a graph G, when G is a tree and a 2-edge connected graph, all the vertices of which have their degrees equal to 2, 3, 6, or 11 modulo 12, and either G can be imbedded with triangular faces only or G is a bipartite graph which can be imbedded with squares only as faces. In the other cases, we give an upper bound of the genus of line graphs. In this way, we solve the question of the Hamiltonian genus of the complete graph K_n, for every n ≥ 3.     

Triangulations of orientable surfaces by complete tripartite graphs

Orientable triangular embeddings of the complete tripartite graph K_n,n,n correspond to biembeddings of Latin squares. We show that if n is prime there are at least e^{nlnn-n(1+o(1))} nonisomorphic biembeddings of cyclic Latin squares of order n. If n=kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n is at least e^{plnp-p(1+lnk+o(1))}. Moreover, we prove that for every n there is a unique regular triangular embedding of K_n,n,n in an orientable surface.     

On 6-sparse Steiner triple systems

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.     

Orthogonal representations of Steiner triple system incidence graphs

The unique Steiner triple system of order $7$ has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in ${\mathbb{C}}^d$ to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that $d=10$ is the smallest number of dimensions in which such a representation exists, a value known as the minimum semidefinite rank of the graph, and give such a representation in $10$ real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound.     

Steiner pentagon systems

A Steiner pentagon system is a pair (K_n, P) where K_n isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition K_n, and such that every part of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.