Nonorientable Embeddings Research Articles

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.

An Euler‐genus approach to the calculation of the crosscap‐number polynomial

AbstractIn 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial.

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→∞.

Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

AbstractWe construct a face two‐colourable, blue and green say, embedding of the complete graph in a nonorientable surface in which there are blue faces each of which have a hamilton cycle as their facial walk and green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of , for all and . Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph , for where and .

The nonorientable genus of the join of two cycles

In this paper, we show that the nonorientable genus of C m + C n , the join of two cycles C m and C n , is equal to $\left\lceil {\tfrac{{(m - 2)(n - 2)}} {2}} \right\rceil $ if m = 3, n ≡ 1 (mod 2), or m ≥ 4, n ≥ 4, (m, n) ≠ (4, 4). We determine that the nonorientable genus of C 4 +C 4 is 3, and that the nonorientable genus of C 3 +C n is $\tfrac{n} {2} $ if n ≡ 0 (mod 2). Our results show that a minimum nonorientable genus embedding of the complete bipartite graph K m,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.

Embedding Distributions and Chebyshev Polynomials

The history of genus distributions began with J. Gross et al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, which can be solved in terms of Chebyshev polynomials of the second kind. The method here can find explicit formula for embedding distribution of some other graphs. As an application, the well known genus distributions of closed-end ladders and cobblestone paths (Furst et al. in J Combin Ser B 46:22–36, 1989) are derived. The explicit formula for non-orientable embedding distributions of closed-end ladders and cobblestone paths are also obtained.

Total embedding distributions of Ringel ladders

The total embedding distributions of a graph consists of the orientable embeddings and non-orientable embeddings and are known for only a few classes of graphs. The orientable genus distribution of Ringel ladders is determined in [E.H. Tesar, Genus distribution of Ringel ladders, Discrete Mathematics 216 (2000) 235–252] by E.H. Tesar. In this paper, using the overlap matrix, we obtain nonhomogeneous recurrence relation for rank distribution polynomial, which can be solved by the Chebyshev polynomials of the second kind. The explicit formula for the number of non-orientable embeddings of Ringel ladders is obtained. Also, the orientable genus distribution of Ringel ladders is re-derived.

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

AbstractWe prove that for every prime number p and odd m>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>0 for which there are at least z nonisomorphic face 2‐colorable triangular embeddings of Kz. © 2011 Wiley Periodicals, Inc. J Graph Theory

Orientable embeddings and orientable cycle double covers of projective-planar graphs

In a closed 2 - cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2 -connected graph has a closed 2 -cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2 -connected projective-planar cubic graph has a closed 2 -cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4 -cycles for orientable closed 2 -cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2 -edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.

Nonorientable regular embeddings of graphs of order pq

In [J. Algeb. Combin.19(2004), 123–141], Du et al. classified the orientable regular embeddings of connected simple graphs of order pq for any two primes p and q. In this paper, we shall classify the nonorientable regular embeddings of these graphs, where p ≠ q. Our classification depends on the classification of primitive permutation groups of degree p and degree pq but is independent of the classification of the arc-transitive graphs of order pq.

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.

The nonorientable genus of joins of complete graphs with large edgeless graphs

We show that for n = 4 and n ⩾ 6 , K n has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, K m ¯ + K n = K m + n − K m , and show that for n ⩾ 3 and m ⩾ n − 1 its nonorientable genus is ⌈ ( m − 2 ) ( n − 2 ) / 2 ⌉ except when ( m , n ) = ( 4 , 5 ) . We then extend these results to find the nonorientable genus of all graphs K m ¯ + G where m ⩾ | V ( G ) | − 1 . We provide a result that applies in some cases with smaller m when G is disconnected.

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.

Another Proof of the Map Color Theorem for Nonorientable Surfaces

The simplest known proof of the Map Color Theorem for nonorientable surfaces (obtained by Youngs, Ringel et al. and given in Ringel's book “Map Color Theorem”) uses index one and three current graphs, and index two and three inductive constructions. We give another proof, still using current graphs, but simpler than Youngs' and Ringel's in several ways. Our proof uses only index one current graphs and no inductive constructions. For every h⩾8, h≠9, 14, we construct an index one current graph Γ( h) that yields a minimal nonorientable embedding ψ( h) of K h . The current graphs Γ( h) have the property that Γ( n) and Γ( n+1) are not too different from each other and share common ladder-like fragments. As a result, the embeddings ψ( n) and ψ( n+1) have a large number of common faces: it is shown that, as n approaches infinity, for n≢3, 9 mod 12 (resp. n≡3,9 mod 12) no less than 5/8 (resp. 5/16) of all faces of ψ( n) appear in ψ( n+1).

On the orientable genus of graphs with bounded nonorientable genus

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

The medial graph and voltage- current duality

The theories of current graphs and voltage graphs give powerful methods for constructing graph embeddings and branched coverings of surfaces. Gross and Alpert first showed that these two theories were dual, that is, that a current assignment on an embedded graph was equivalent to a voltage assignment on the embedded dual. In this paper we examine current and voltage graphs in the context of the medial graph, a 4-regular graph formed from an embedded graph which encodes both the primal and dual graphs. As a consequence we obtain new insights into voltage-current duality, including wrapped coverings. We also develop a method for simultaneously giving a voltage and a current assignment on an embedded graph in the case that the voltage-current group is abelian. We apply this technique to construct self-dual embeddings for a variety of graphs. We also construct orientable and non-orientable embeddings of K p, q with dual K r, s for all possible p, q, r, s even with pq = rs.

Characterization of the maximum genus of a signed graph

A signed graph is a graph in which each edge is labelled with a sign. A cycle C of a signed graph is balanced if the product of signs on C is positive. By an embedding of a signed graph G on a closed surface S we mean a topological embedding of the corresponding unsigned graph satisfying the following additional condition: A cycle C of G is embedded as an orientation-preserving cycle on S if and only if C is balanced. Let ε M( G) be the maximum Euler genus of a closed surface in which the signed graph G has a 2-cell embedding. We establish a minimax formula for ε M( G). The method used provides a unification of several results in maximum genus of unsigned graphs. For example, Xuong's and Nebeský's formulas for the maximum orientable genus of a graph and the Ringel-Stahl singleface nonorientable embedding theorem are corollaries of the work in this paper.

Regular embeddings of coloured digraphs

A particular kind of 2-cell embeddings, called regular, for arc-coloured digraphs is introduced, and a method for constructing both orientable and non-orientable regular embeddings is presented. Furthermore, by using combinatorial concepts and the Euler-Poincare formula, we derive upper bounds for both the orientable and non-orientable genera of such arc-coloured digraphs.

Self-dual embeddings of Cayley graphs

A method for constructing both orientable and non-orientable embeddings of Cayley graphs is described. This method yields all the previously known constructions in addition to producing some new self-dual embeddings.