Nonorientable biembeddings of Steiner triple systems

Constructions due to Ringel show that there exists a nonorientable face 2-colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n≡3 (mod 6) with n9. We prove the corresponding existence theorem for n≡1 (mod 6) with n13

On the bi-embeddability of certain Steiner triple systems of order 15

There are 80 non-isomorphic Steiner triple systems of order 15. A standard listing of these is given in Mathon et al.(1983, Ars Combin., 15, 3–110). We prove that systems #1 and #2 have no bi-embedding together in an orientable surface. This is the first known example of a pair of Steiner triple systems of ordern , satisfying the admissibility condition n ≡ 3 or 7(mod 12), which admits no orientable bi-embedding. We also show that the same pair has five non-isomorphic bi-embeddings in a non-orientable surface

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 that generates an orientable face 2-colorable triangular embedding of the complete graph K12n+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

A lower bound for the number of orientable triangular embeddings of some complete graphs

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic face 2-colourable triangular embeddings of the complete graph Kn in an orientable surface is at least nan2

Exponential families of non-isomorphic triangulations of complete graphs

We prove that the number of non-isomorphic face 2-colourable triangulations of the complete graph $K_n$ in an orientable surface is at least $2^{n^2/54 -O(n)}$ for $n$ congruent to 7 or 19 modulo 36, and is at least $2^{2n^2/81 -O(n)}$ for $n$ congruent to 19 or 55 modulo 108

Pasch trades with a negative block

A Steiner triple system of order $v$, STS($v$), may be called \emph{equivalent} to another STS($v$) if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS($v$)s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system $S$ on a base set $V$ contains all the systems that can be obtained from $S$ by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of $S$ on $V$. We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block

Modular gracious labellings of trees

A gracious labelling g of a tree is a graceful labelling in which, treating the tree as a bipartite graph, the label of any edge (d,u) (d a 'down' and u an 'up' vertex) is g(u) - g(d). A gracious k-labelling is one such that each residue class modulo k has teh 'correct' numbers of vertex and edge labels -- that is, the numbers that arise by interpreting the labels of a gracious labelling modulo k. In this paper it is shown that every non-null tree has a gracious k-labelling for each k = 2,3,4,5

On colourings of Steiner triple systems

A Steiner triple system, STS(v), is said to be χ-chromatic if the points can be coloured using χ colours, but no fewer, such that no block is monochromatic. All known 3-chromatic STS(v) are also equitably colourable, i.e. there exists a 3-colouring in which the cardinalities of the colour classes differ by at most one. We present examples of 3-chromatic STS(v) which do not admit equitable 3-colourings. We also present further examples of systems with unique and balanced colouring

Biembeddings of Latin squares and Hamiltonian decompositions

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ and 6

The Triangle chromatic index of Steiner triple systems

In a Steiner triple system of order v, STS(v), a set of three lines intersecting pairwise in three distinct points is called a triangle. A set of lines containing no triangle is called triangle-free. The minimum number of triangle-free sets required to partition the lines of a Steiner triple system S, is called the triangle chromatic index of S. We prove that for all admissible v, there exists an STS (v) with triangle chromatic index at most 8√3v. In addition, by showing that the projective geometry PG(n,3) may be partitioned into O(6n/5) caps, we prove that the STS(v) formed the points and lines of the affine geometry AG(n,3) has triangle chromatic index at most Avs, where s=log6/(3log5)≈0.326186, and A is a constant. We also determine the values of the index for STS(v) with v≤13