A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic triangular embeddings of the complete graph K_n is at least n^an2. A similar lower bound is also given, for an infinite set of values of n, on the number of nonisomorphic triangular embeddings of the complete regular tripartite graph K_n,n,n.     

A decomposition of (<Emphasis Type="Italic">λK</Emphasis><Subscript><Emphasis Type="Italic">υ</Emphasis></Subscript>)<Superscript>+</Superscript> with extended triangles

By an extended triangle, we mean a loop, a loop with an edge attached (known as a lollipop), or a copy of K ₃ (known as a triangle). In this paper, we completely solve the problem of decomposing the graph (λK _υ )⁺ into extended triangles for all possible number of loops. Dedicated to Professor Alex Rosa on the occasion of his 70th birthday     

On semi-planar Steiner quasigroups

A Steiner triple system (briefly ST) is in 1-1 correspondence with a Steiner quasigroup or squag (briefly SQ) [B. Ganter, H. Werner, Co-ordinatizing Steiner systems, Ann. Discrete Math. 7 (1980) 3-24; C.C. Lindner, A. Rosa, Steiner quadruple systems: A survey, Discrete Math. 21 (1979) 147-181]. It is well known that for each n≡1 or 3 (mod 6) there is a planar squag of cardinality n [J. Doyen, Sur la structure de certains systems triples de Steiner, Math. Z. 111 (1969) 289-300]. Quackenbush expected that there should also be semi-planar squags [R.W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28 (1976) 1187-1198]. A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element squag. The first author has constructed a semi-planar squag of cardinality 3n for all n>3 and n≡1 or 3 (mod 6) [M.H. Armanious, Semi-planar Steiner quasigroups of cardinality 3n, Australas. J. Combin. 27 (2003) 13-27]. In fact, this construction supplies us with semi-planar squags having only nontrivial subsquags of cardinality 9. Our aim in this article is to give a recursive construction as n→3n for semi-planar squags. This construction permits us to construct semi-planar squags having nontrivial subsquags of cardinality >9. Consequently, we may say that there are semi-planar (or semi-planar ) for each positive integer m and each n≡1 or 3 (mod 6) with n>3 having only medial subsquags at most of cardinality 3^ν (sub-) for each ν∈{1,2,…,m+1}.     

Nonorientable triangular embeddings of complete graphs with arbitrarily large looseness

The looseness of a triangular embedding of a complete graph in a closed surface is the minimum integer m such that for every assignment of m colors to the vertices of the embedding (such that all m colors are used) there is a face incident with vertices of three distinct colors. In this paper we show that for every p?3 there is a nonorientable triangular embedding of a complete graph with looseness at least p.     

Completing partial commutative quasigroups constructed from partial Steiner triple systems is NP-complete

Deciding whether an arbitrary partial commutative quasigroup can be completed is known to be NP-complete. Here, we prove that it remains NP-complete even if the partial quasigroup is constructed, in the standard way, from a partial Steiner triple system. This answers a question raised by Rosa in [A. Rosa, On a class of completable partial edge-colourings, Discrete Appl. Math. 35 (1992) 293-299]. To obtain this result, we prove necessary and sufficient conditions for the existence of a partial Steiner triple system of odd order having a leave L such that E(L)=E(G) where G is any given graph.     

Orientable biembeddings of cyclic Steiner triple systems from current assignments on Möbius ladder graphs

We give a characterization of a current assignment on the bipartite Möbius ladder graph with 2n+1 rungs. Such an assignment yields an index one current graph with current group Z_12n+7 that generates an orientable face 2-colorable triangular embedding of the complete graph K_12n+7 or, equivalently, an orientable biembedding of two cyclic Steiner triple systems of order 12n+7. We use our characterization to construct Skolem sequences that give rise to such current assignments. These produce many nonisomorphic orientable biembeddings of cyclic Steiner triple systems of order 12n+7.     

The spectrum for quasigroups with cyclic automorphisms and additional symmetries

We determine necessary and sufficient conditions for the existence of a quasigroup of order n having an automorphism consisting of a single cycle of length m and n−m fixed points, and having any combination of the additional properties of being idempotent, unipotent, commutative, semi-symmetric or totally symmetric. Quasigroups with such additional properties and symmetries are equivalent to various classes of triple systems.     

Groupoids with quasigroup and Latin square properties

Connections of some groupoid identities with the quasigroup and Latin square properties are investigated using combinatorial methods.     

Colouring Cubic Graphs by Small Steiner Triple Systems

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]). Research partially supported by APVT, project 51-027604. Research partially supported by VEGA, grant 1/3022/06.