Orientable Embeddings Research Articles

The connectivity of the dual

AbstractThe dual of a polyhedron is a polyhedron—or in graph‐theoretical terms: the dual of a 3‐connected plane graph is a 3‐connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity of the dual on higher surfaces. Are the duals of 3‐connected embedded graphs of higher genus 3‐connected, too? If not, which connectivity guarantees 3‐connectedness of the dual? In this article, we give answers to some of these and related questions. We prove that there is no connectivity that guarantees the 3‐connectedness or 2‐connectedness of the dual for every genus, and give upper bounds for the minimum genus for which (with ) a c‐connected embedded graph with a dual that has a 1‐ or 2‐cut can occur. We prove that already on the torus, we need 6‐connectedness to guarantee 3‐connectedness of the dual and 4‐connectedness to guarantee 2‐connectedness of the dual. In the last section, we answer a related question by Plummer and Zha on orientable embeddings of highly connected noncomplete graphs.

Recurrences for the genus polynomials of linear sequences of graphs

Abstract Given a finite graph H, the n th member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G n−1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ G (z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ Gn (z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials Γ G n j $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector V n ( z ) = [ Γ G n 1 ( z ) , Γ G n 2 ( z ) , … , Γ G n k ( z ) ] t $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion Vn (z) = M(z)V n−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k th degree linear recursion for Γ n (z), allowing us to avoid the partitioning, thereby yielding a reduction from k 2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Quantum walks on embeddings

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of \(U^2\), and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

Construction of quantum codes from new embeddings of graphs on compact surfaces

In this paper we introduce a new class of sparse CSS quantum error-correcting codes based on new orientable self-dual embeddings of graphs. Each code in this class is associated with the embedding of the surface so that the qubits correspond to the edges of the embedding. The parameters of the new codes are $$[[\frac{m(m+1)}{2},\frac{m(m-3)}{2},3]]$$, where $$m=4s$$, $$s\ge 1$$. We also present a table of quantum codes whose parameters had not been shown before.

Classification of some reflexible edge-transitive embeddings of complete bipartite graphs

In this paper, we classify some reflexible edge-transitive orientable embeddings of complete bipartite graphs. As a by-product, we classify groups Γ such that (i) Γ = XY for some cyclic groups X = ⟨ x ⟩ and Y = ⟨ y ⟩ with X ∩ Y = {1 Γ } and (ii) there exists an automorphism of Γ which sends x and y to x −1 and y −1 , respectively.

Edge-outer graph embedding and the complexity of the DNA reporter strand problem

In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a DNA reporter strand, that is, a closed walk covering every edge either once or twice, in opposite directions if twice, and passing through each vertex in a particular way. This corresponds to showing that every graph has an edge-outer embedding, that is, an orientable embedding with some face that is incident with every edge. In the motivating application, the objective is such a closed walk of minimum length. Here we give a short algorithmic proof of the original existence result, and also prove that finding a shortest length solution is NP-hard, even for 3-connected cubic (3-regular) planar graphs. Independent of the motivating application, this problem opens a new direction in the study of graph embeddings, and we suggest several problems emerging from it.

New classes of quantum codes on closed orientable surfaces

In this paper we construct two classes of binary quantum error-correcting codes on closed orientable surfaces. These codes are derived from self-dual orientable embeddings of complete bipartite graphs and complete multipartite graphs on the corresponding closed orientable surfaces. We also show a table comparing the rate of these quantum codes when fixing the minimum distance to 3 and 4.

Classes of quantum codes derived from self-dual orientable embeddings of complete multipartite graphs

This paper presents four classes of binary quantum codes with minimum distance 3 and 4, namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the complete graphs $$K_{4r+1}$$ and $$K_{4s}$$ and by current graphs and rotation schemes. The parameters of two classes of quantum codes are $$[[2r(4r+1),2r(4r-3),3]]$$ and $$[[2s(4s-1),2(s-1)(4s-1),3]]$$ , respectively, where $$r\ge 1$$ and $$s\ge 2$$ . For these quantum codes, the code rate approaches 1 as r and s tend to infinity. The Class-III with minimum distance 4 is constructed by using self-dual embeddings of complete bipartite graphs. The parameters of this class are $$\left[ \left[ rs,\frac{(r-2)(s-2)}{2},4\right] \right] $$ , where r and s are both divisible by 4. The proposed Class-IV is the minimum distance 3 and code length $$n=(2r+1)s^{2}$$ . This class is constructed based on self-dual embeddings of complete tripartite graph $$K_{rs,s,s}$$ , and its parameters are $$[[(2r+1)s^{2},(rs-2)(s-1),3]]$$ , where $$r\ge 2$$ and $$s\ge 2$$ .

Odd components of co-trees and graph embeddings

In this paper we investigate the relation between odd components of co-trees and graph embeddings. We show that any graph G must share one of the following two conditions: (a) for each integer h such that G may be embedded on Sh, the sphere with h handles, there is a spanning tree T in G such that h=12(β(G)−ω(T)), where β(G) and ω(T) are, respectively, the Betti number of G and the number of components of G−E(T) having odd number of edges; (b) for every spanning tree T of G, there is an orientable embedding of G with exact ω(T)+1 faces. This extends Xuong and Liu’s theorem [9,13] to some other (possible) genera. Infinitely many examples show that there are graphs which satisfy (a) but (b). Those make a correction of a result of Archdeacon [2, Theorem 1].

Topological quantum codes from self-complementary self-dual graphs

In this paper, we present two new classes of binary quantum codes with minimum distance of at least three, by self-complementary self-dual orientable embeddings of voltage and Paley graphs in the Galois field $$GF(p^{r})$$GF(pr), where $$p\in {\mathbb {P}}$$p?P and $$r\in {\mathbb {Z}}^{+}$$r?Z+. The parameters of two new classes of quantum codes are $$[[(2k'+2)(8k'+7),2(8k'^{2}+7k'),d_\mathrm{min}]]$$[[(2k?+2)(8k?+7),2(8k?2+7k?),dmin]] and $$[[(2k'+2)(8k'+9),2(8k'^{2}+9k'+1),d_\mathrm{min}]]$$[[(2k?+2)(8k?+9),2(8k?2+9k?+1),dmin]], respectively, where $$d_\mathrm{min}\ge 3$$dmin?3. For these quantum codes, the code rate approaches 1 as $$k'$$k? tends to infinity.

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.

Counting Orientable Embeddings by Genus for a Type of 3-Regular Graph

On the basis of the joint tree model initiated in Liu (The non-orientable maximum genus of a graph (in Chinese), Scieulia Sinica, Special Issue on Math. I 191–201, 1979) and comprehensively described in Liu (Theory of Polyhedra (in English), Science Press, Beijing, 2008), this paper provides the numbers of topologically non-equivalent orientable embeddings of a new type of 3-regular graphs by genus via classifying the associate polyhegons.

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.

Regular embeddings of [formula omitted] where [formula omitted] is a power of 2. II: The non-metacyclic case

The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K n , n , where n = 2 e . The method involves groups G which factorise as a product G = X Y of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n = 2 e , e ≥ 3 , there are up to map isomorphism exactly four regular embeddings of K n , n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n = 4 .

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say n = 2 a p 1 a 1 p 2 a 2 ⋯ p k a k ( p 1 , p 2 , … , p k are distinct odd primes and a i > 0 for each i ⩾ 1 ) , there are t distinct reflexible regular embeddings of the complete bipartite graph K n , n up to isomorphism, where t = 1 if a = 0 , t = 2 k if a = 1 , t = 2 k + 1 if a = 2 , and t = 3 · 2 k + 1 if a ⩾ 3 . And, there are s distinct self-Petrie dual regular embeddings of K n , n up to isomorphism, where s = 1 if a = 0 , s = 2 k if a = 1 , s = 2 k + 1 if a = 2 , and s = 2 k + 2 if a ⩾ 3 .

Orientable embedding genus distribution for certain types of graphs

In this paper, on the bases of joint trees for a graph introduced by Liu, a surface generating method is developed to determine embedding genus distribution of a graph.

Regular orientable embeddings of complete bipartite graphs

AbstractIn this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph Kn,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 Kn,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐Škoviera's question raised in 12. © 2005 Wiley Periodicals, Inc. J Graph Theory

On Essential and Inessential Polygons in Embedded Graphs

In this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it has a face chain containing a non-contractible closed path. Our main application is to prove that any two faces of a 4-representative embedding are simultaneously contained in a disc bounded by a polygon. This result is used to prove the existence of ⌊(r−1)/8⌋ pairwise disjoint, pairwise homotopic non-contractible separating polygons in an r -representative orientable embedding. Our proof of this latter result is simple and mechanical.