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.     

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.     

Lower Bound of the Number of Maximum Genus Embeddings and Genus Embeddings of K 12s+7

In this paper, we study lower bound of the number of maximum orientable genus embeddings (or MGE in short) for a loopless graph. We show that a connected loopless graph of order n has at least \frac14^g_M(G)?_{v ? V(G)}(d(v)-1)!{\frac{1}{4^{\gamma_M(G)}}\prod_{v\in{V(G)}}{(d(v)-1)!}} distinct MGE’s, where γ _M (G) is the maximum orientable genus of G. Infinitely many examples show that this bound is sharp (i.e., best possible) for some types of graphs. Compared with a lower bound of Stahl (Eur J Combin 13:119–126, 1991) which concerns upper-embeddable graphs (i.e., embedded graphs with at most two facial walks), this result is more general and effective in the case of (sparse) graphs permitting relative small-degree vertices. We also obtain a similar formula for maximum nonorientable genus embeddings for general graphs. If we apply our orientable results to the current graph G _s of K _12s+7, then G _s has at least 2^3s distinct MGE’s.This implies that K _12s+7 has at least (22) ^s nonisomorphic cyclic oriented triangular embeddings for sufficient large s.     

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.     

Voltage graph embeddings and the associated block designs

The voltage graph construction of Gross (orientable case) and Stahl as well as Gross and Tucker (nonorientable case) is extended to the case where the base graph is embedded in a pseudosurface or a generalized pseudosurface. This theory is then applied to produce triangular embeddings of K_4(n); they in turn yield an infinite class of partially balanced incomplete block designs.     

Recursive constructions for triangulations

Three recursive constructions are presented; two deal with embeddings of complete graphs and one with embeddings of complete tripartite graphs. All three facilitate the construction of 2) non‐isomorphic face 2‐colourable triangulations of K_n and K_n,n,n in orientable and non‐orientable surfaces for values of n lying in certain residue classes and for appropriate constants a. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 87–107, 2002     

Self-dual embeddings of complete multipartite graphs

In this paper we examine self-dual embeddings of complete multipartite graphs, focusing primarily on K_m(n) having m parts each of size n. If m = 2, then n must be even. If the embedding is on an orientable surface, then an Euler characteristic argument shows that no such embedding exists when n is odd and m ? 2, 3 (mod 4); there is no such restriction for embeddings on nonorientable surfaces. We show that these embeddings exist with a few small exceptions. As a corollary, every group has a Cayley graph with a self-dual embedding. Our main technique is an addition construction that combines self-dual embeddings of two subgraphs into a self-dual embedding of their union. We also apply this technique to nonregular multipartite graphs and to cubes.     

On the number of triangular embeddings of complete graphs and complete tripartite graphs

We prove that for every prime number p and odd m>1, as s→∞, there are at least w face 2‐colorable triangular embeddings of K_{w, w, w}, where w = m·p^s. For both orientable and nonorientable embeddings, this result implies that for infinitely many infinite families of z, there is a constant c>0 for which there are at least z nonisomorphic face 2‐colorable triangular embeddings of K_z. © 2011 Wiley Periodicals, Inc. J Graph Theory     

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.     

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     

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     

Regular orientable embeddings of complete bipartite graphs

In this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph K_n,n are in one‐to‐one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of K_n,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐?koviera's question raised in 12 . © 2005 Wiley Periodicals, Inc. J Graph Theory     

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.     

Dihedral Biembeddings and Triangulations by Complete and Complete Tripartite Graphs

We construct biembeddings of some Latin squares which are Cayley tables of dihedral groups. These facilitate the construction of ${n^{an^2}}$ nonisomorphic face 2-colourable triangular embeddings of the complete tripartite graph K _n,n,n and the complete graph K _n for linear classes of values of n and suitable constants a. Previously the best known lower bounds for the number of such embeddings that are applicable to linear classes of values of n were of the form ${2^{an^2}.}$ We remark that trivial upper bounds are ${n^{n^2/3}}$ in the case of complete graphs K _n and ${n^{2n^2}}$ in the case of complete tripartite graphs K _n,n,n .     

Classification of Regular Embeddings of Complete Multipartite Graphs

A 2‐cell embedding of a graph Γ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags. In this article, we classify the regular embeddings of the complete multipartite graph $K_{n,\dots ,n}$. We show that if the number of partite sets is greater than 3, there exists no such embedding; and if the number of partite sets is 3, for any n, there exist one orientable regular embedding and one nonorientable regular embedding of $K_{n,n,n}$ up to isomorphism.     

Quadrilateral embeddings of bipartite graphs

Current graphs and a theorem of White are used to show the existence of almost complete regular bipartite graphs with quadrilateral embeddings conjectured by Pisanski. Decompositions of K_n and K_{n, n} into graphs with quadrilateral embeddings are discussed, and some thickness results are obtained. Some new genus results are also obtained.     

Cochromatic Number and the Genus of a Graph

The cochromatic number of a graph G, denoted by z(G), is the minimum number of subsets into which the vertex set of G can be partitioned so that each sbuset induces an empty or a complete subgraph of G. In this paper we introduce the problem of determining for a surface S, z(S), which is the maximum cochromatic number among all graphs G that embed in S. Some general bounds are obtained; for example, it is shown that if S is orientable of genus at least one, or if S is nonorientable of genus at least four, then z(S) is nonorientable of genus at least four, then z(S)≤χ(S). Here χ(S) denotes the chromatic number S. Exact results are obtained for the sphere, the Klein bottle, and for S. It is conjectured that z(S) is equal to the maximum n for which the graph G_n = K₁ ∪ K₂ ∪ … ∪ K_n embeds in S.     

Index four orientable embeddings and case zero of the Heawood conjecture

By imposing a special symmetry, we are able to construct index four triangular embeddings of graphs in compact orientable 2-manifolds. Because of the complexity of the current graphs required, such embeddings have heretofore been unattainable, but the imposed symmetry reduces the problem to constructing a special kind of index two current graph. We illustrate the method with a solution for case zero of the Heawood conjecture, using an abelian group, thus completing a constructive proof of the Heawood map color theorem, and eliminating the need for Galois field theory and nonabelian groups in its solution. The method has also been used in the determination of the genus of K_n,n,n,n.     

Maximum genus embeddings of Steiner triple systems

We prove that for n>3 every STS(n) has both an orientable and a nonorientable embedding in which the triples of the STS(n) appear as triangular faces and there is just one additional large face. We also obtain detailed results about the possible automorphisms of such embeddings.     

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.