Triangular Embeddings Research Articles

Index 3 biembeddings of the complete graphs

We show that the complete graphs on 24s+21 vertices have decompositions into two edge-disjoint subgraphs, each of which triangulates an orientable surface. The special case where the two surfaces are homeomorphic solves a generalized Earth-Moon problem for that surface. Unlike previous constructions, these pairs of triangular embeddings are derived from index 3 current graphs.

Design and Analysis of Systematic Batched Network Codes.

Systematic codes are of important practical interest for communications. Network coding, however, seems to conflict with systematic codes: although the source node can transmit message packets, network coding at the intermediate network nodes may significantly reduce the number of message packets received by the destination node. Is it possible to obtain the benefit of network coding while preserving some properties of the systematic codes? In this paper, we study the systematic design of batched network coding, which is a general network coding framework that includes random linear network coding as a special case. A batched network code has an outer code and an inner code, where the latter is formed by linear network coding. A systematic batched network code must take both the outer code and the inner code into consideration. Based on the outer code of a BATS code, which is a matrix-generalized fountain code, we propose a general systematic outer code construction that achieves a low encoding/decoding computation cost. To further reduce the number of random trials required to search a code with a close-to-optimal coding overhead, a triangular embedding approach is proposed for the construction of the systematic batches. We introduce new inner codes that provide protection for the systematic batches during transmission and show that it is possible to significantly increase the expected number of message packets in a received batch at the destination node, without harm to the expected rank of the batch transfer matrix generated by network coding.

A simple proof of the Map Color Theorem for nonorientable surfaces

We give a more simple proof of the Map Color Theorem for nonorientable surfaces that uses only four constructions of current graphs instead of 12 constructions used in the previous proof. For every i=0,1,2,3, using an index one current graph with cyclic current group, we construct a nonorientable triangular embedding of K12s+3i+1 that can be easily modified into a nonorientable triangular embedding of K12s+3i+3 and a minimal nonorientable embedding of K12s+3i+5. As a result, for every n≥10, n∉{11,14,20}, we construct a minimal nonorientable embedding of Kn. During the modifications, all but 2(4s+i) faces of the nonorientable triangular embedding of K12s+3i+1 become faces of the nonorientable triangular embedding of K12s+3i+3, and all but 2(4s+i)+1 faces of the nonorientable triangular embedding of K12s+3i+3 become faces of the minimal nonorientable embedding of K12s+3i+5.

A simple construction of exponentially many nonisomorphic orientable triangular embeddings of K_12s

Using an index one current graph with the cyclic current group we give a simple construction of 22s − 7 nonisomorphic orientable triangular embeddings of the complete graph K12s, s ≥ 4. These embeddings have no nontrivial automorphisms.

A simple construction for orientable triangular embeddings of the complete graphs on [formula omitted] vertices

We present a family of index 1 abelian current graphs whose derived embeddings can be modified into triangular embeddings of K12s for s≥3. Our construction is significantly simpler than previous methods for finding genus embeddings of K12s, which utilized either large index or nonabelian groups.

On the asymptotic growth of bipartite graceful permutations

We give exact growth rates for the number of bipartite graceful permutations of the symbols {0,1,…,n−1} that start with a for a≤14 (equivalently, α-labelings of paths with n vertices that have a as a pendant label). In particular, when a=14 the growth is asymptotically like λn for λ≈3.461. The number of graceful permutations of length n grows at least this fast, improving on the best existing asymptotic lower bound of 2.37n. Combined with existing theory, this improves the known lower bounds on the number of Hamiltonian decompositions of the complete graph K2n+1 and on the number of cyclic oriented triangular embeddings of K12s+3 and K12s+7. We also give the first exponential lower bound on the number of R-sequencings of Z2n+1.

Notes on Prime Numbers, Their Numerical Statistics & Patterns II: Modular Arithmetic and the Symmetric Triangular Embedding Model

Recently we discussed in a letter the modular arithmetic and a particular integer numbers (odd) model, which we then utilized to discuss parsed integer numbers statistics and nearness. We continue in this letter to discuss the approach as applied to other models, namely the triangular embedding of symmetric copies in large number representations, and obtain a relationship for identifying primes from non primes. We then discuss other models that may prove useful such as the golden ratio spiral and number sequences from the same approach.

Auxiliary embeddings and constructing triangular embeddings of joins of complete graphs with edgeless graphs

Until recently there were no results on the orientable or nonorientable genus of graphs Kn+Km¯ for n/2≤m<n−1. Recently, using index one current graphs, McCourt (2014) constructed a nonorientable triangular embedding of K36t+3+K18t+1+6i¯ for i=0,1,…,t−1 and t≥1. In the present paper, using index three current graphs, we construct auxiliary embeddings that allow three arbitrary triangular embeddings of Kn+1 to be incorporated into a triangular embedding of K3n+Km¯ for many cases where m can lie anywhere in the interval 1≤m≤2n. Using these auxiliary embeddings we construct the following six classes of triangular embeddings: orientable triangular embeddings of K36t+6+K1+2i¯ for t≥1 and i=0,1,…,12t, K36t+18+K1+2i¯ for t≥0 and i=0,1,…,12t+4, and nonorientable triangular embeddings of K18t+K1+2i¯ for t≥1 and i=0,1,…,6t−2, K18t+6+K1+2i¯ for t≥1 and i=0,1,…,6t, K18t−3+Ki¯ for t≥2 and i=3,4,…,12t−6, K18t+9+Ki¯ for t≥1 and i=3,4,…,12t+2. Using these auxiliary embeddings we also show that for a certain positive constant a and for an infinite number of values of t, the number of nonisomorphic orientable triangular embeddings of K36t+6+K1+2i¯ for t≥1 and every i∈{0,1,…,12t} is at least (36t+6)a(36t+6)2−o((36t+6)2) as t→∞. Analogous results are obtained for the other five classes of triangular embeddings as well.

Recursive constructions and nonisomorphic minimal nonorientable embeddings of complete graphs

We construct a family of recursive constructions such that for any i∈{0,1,3,4,6,7,9,10} and j∈{0,1,…,11}, several arbitrary nonorientable triangular embeddings of every complete graph Km, m≡i(mod12), can be incorporated into a minimal nonorientable embedding of Km̄, m̄≡j(mod12). The existence of such recursive constructions implies the following important interdependency of the sets of nonisomorphic minimal nonorientable embeddings of Kn for different residue classes of n modulo 12: if for some i∈{0,1,3,4,6,7,9,10}, the number of nonisomorphic nonorientable triangular embeddings of a graph Km, m≡i(mod12), is large enough, then for any other j∈{0,1,…,11}, the number of nonisomorphic minimal nonorientable embeddings of some graph Km̄, m̄≡j(mod12), is also large enough. As a consequence, using Grannell and Knor’s (2013) results for Kn, n≡1 or 9(mod12), we show that there is a certain positive constant a such that for any i∈{0,1,…,11}, there is an infinite set (a linear class) of values of n, where n≡i(mod12), such that the number of nonisomorphic minimal nonorientable embeddings of Kn is at least nan2−o(n2) as n→∞.

Nonorientable biembeddings of cyclic Steiner triple systems generated by Skolem sequences

We describe a class of Skolem sequences of order n such that for the cyclic Steiner triple system of order 6n+1 generated by any Skolem sequence from the class we can construct a bipartite index one current graph generating a nonorientable face 2-colorable triangular embedding of the complete graph K6n+1 such that the triangular faces of one of the two color classes form the blocks of the cyclic Steiner triple system.

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 .

Tuning the band gap and magnetic properties of BN sheets impregnated with graphene flakes

The BN sheet is a nonmagnetic wide-band-gap semiconductor. Using density functional theory, we show that these properties can be fundamentally altered by embedding graphene flakes. Not only do graphene flakes preserve the two-dimensional (2D) planar structure of the BN sheet, but by controlling their shape and size, unexpected electronic and magnetic properties also emerge. The electronic band structure can be tuned from a direct gap to an indirect gap, the energy gap can be further modulated by changing the bonding patterns, and both hole injecting or electron injecting can be achieved by tailoring the triangular embedding pattern. Furthermore, the Lieb theorem still holds, and the embedded triangular graphene flakes become ferromagnetic with full spin polarizations of the introduced electrons or holes, opening the door to their use as spin filters. The study sheds new light on hybrid single-atomic-layer engineering for unprecedented applications of 2D nanomaterials.

A Construction for Biembeddings of Latin Squares

An existing construction for face 2-colourable triangular embeddings of complete regular tripartite graphs is extended and then re-examined from the viewpoint of the underlying Latin squares. We prove that this generalization gives embeddings which are not isomorphic to any of those produced by the original construction.

Biembedding Abelian groups with mates having transversals

A certain recursive construction for biembeddings of Latin squares has played a substantial role in generating large numbers of nonisomorphic triangular embeddings of complete graphs. In this article, we prove that, except for the groups and C4, each Latin square formed from the Cayley table of an Abelian group appears in a biembedding in which the second Latin square has a transversal. Such biembeddings may then be freely used as ingredients in the recursive construction. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:81-88, 2012

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

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

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 \({\frac{1}{4^{\gamma_M(G)}}\prod_{v\in{V(G)}}{(d(v)-1)!}}\) distinct MGE’s, where γM(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 Gs of K12s+7, then Gs has at least 23s distinct MGE’s.This implies that K12s+7 has at least (22)s nonisomorphic cyclic oriented triangular embeddings for sufficient large s.

THIRD-REGULAR BI-EMBEDDINGS OF LATIN SQUARES

AbstractFor each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of Kn, n, and this generates a regular face 2-colourable triangular embedding of Kn, n, n. In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of Kn, n. This paper presents an analysis of the face 2-colourable triangular embedding of Kn, n, n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.

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 K n in an orientable surface is at least n a n 2 .

On Biembeddings of Latin Squares

A known construction for face 2-colourable triangular embeddings of complete regular tripartite graphs is re-examined from the viewpoint of the underlying Latin squares. This facilitates biembeddings of a wide variety of Latin squares, including those formed from the Cayley tables of the elementary Abelian 2-groups $C_2^k$ ($k\ne 2$). In turn, these biembeddings enable us to increase the best known lower bound for the number of face 2-colourable triangular embeddings of $K_{n,n,n}$ for an infinite class of values of $n$.

Biembeddings of Abelian groups

AbstractWe prove that, with the single exception of the 2‐group C, the Cayley table of each Abelian group appears in a face 2‐colorable triangular embedding of a complete regular tripartite graph in an orientable surface. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 71–83, 2010