Orientable Genus Research Articles

Finite Groups Whose Co-prime Order Graphs Have Positive Genus

Given a finite group G, the co-prime order graph of G is the simple undirected graph whose vertex set is G, and two distinct vertices x; y are adjacent if gcd(o(x); o(y)) = 1 or p, where p is a prime, and o(x) and o(y) are the orders of x and y, respectively. In this paper, we prove that, for a fixed positive integer k, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus k. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.

Partial-dual Euler-genus distributions for bouquets with small Euler genus

For an arbitrary ribbon graph G, the partial-dual Euler-genus polynomial of G is a generating function that enumerates partial duals of G by Euler genus. When G is an orientable ribbon graph, the partial-dual orientable genus polynomial of G is a generating function that enumerates partial duals of G by orientable genus. Gross, Mansour, and Tucker inaugurated these partial-dual Euler-genus and orientable genus distribution problems in 2020. A bouquet is a one-vertex ribbon graph. Given a ribbon graph G, its partial-dual Euler-genus polynomial is the same as that of some bouquet; this motivates our focus on bouquets. We obtain the partial-dual Euler-genus polynomials for all the bouquets with Euler genus at most two.

Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an analogous conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In this paper, we first give some counterexamples to the conjecture. Then motivated by these counterexamples, we further find two infinite classes of counterexamples.

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem . Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O (log n / log log n ) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

Generalisations of the Harer–Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer–Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to Bousquet-Mélou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

Counterexamples to a conjecture by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials for all ribbon graphs with the number of edges less than 4 and the partial-dual orientable genus polynomials for all orientable ribbon graphs with the number of edges less than 5 in terms of signed interlace sequences. We check all the conjectures and find a counterexample to the Conjecture 3.1 in their paper: There is no orientable ribbon graph having a non-constant partial-dual genus polynomial with only one non-zero coefficient. Motivated by this counterexample, we further find an infinite family of counterexamples to the conjecture.

Finite groups whose noncyclic graphs have positive genus

For a finite noncyclic group G, let $${\rm Cyc} (G)$$ be a set of elements a of G such that $$\langle a, b\rangle$$ is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set $$G\setminus {\rm Cyc} (G)$$ , having an edge between two distinct vertices x and y if $$\langle x, y\rangle$$ is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively.

The orientable genus of the join of a cycle and a complete graph

Let m and n be two integers. In the paper we show that the orientable genus of the join of a cycle C m and a complete graph K n is ⌈( m − 2)( n − 2) / 4⌉ if n = 4 and m ≥ 12 , or n ≥ 5 and m ≥ 6 n − 13 .

Power graphs of (non)orientable genus two

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this article, we classify the finite groups whose power graphs have (non)orientable genus two.

Genus of the Cartesian Product of Triangles.

We investigate the orientable genus of $G_n$, the cartesian product of $n$ triangles, with a particular attention paid to the two smallest unsolved cases $n=4$ and $5$. Using a lifting method we present a general construction of a low-genus embedding of $G_n$ using a low-genus embedding of $G_{n-1}$. Combining this method with a computer search and a careful analysis of face structure we show that $30\le \gamma(G_4) \le 37$ and $133 \le\gamma(G_5) \le 190$. Moreover, our computer search resulted in more than $1300$ non-isomorphic minimum-genus embeddings of $G_3$. We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of $Z_p^n$ is calculated for all odd primes $p$.

Genus Ranges of 4-Regular Rigid Vertex Graphs

A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with 2n vertices (n > 1), we prove that all intervals [a, b] for all a < b ≤ n, and singletons [h, h] for some h ≤ n, are realized as genus ranges. For graphs with 2n - 1 vertices (n ≥ 1), we prove that all intervals [a, b] for all a < b ≤ n except [0, n], and [h, h] for some h ≤ n, are realized as genus ranges. We also provide constructions of graphs that realize these ranges.

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum-size matching and it is factor critical if G-v has a perfect matching for each vertex v of G. It is known that any 2-connected equimatchable graph is either bipartite or factor critical. We prove that for 2-connected factor-critical equimatchable graph G the graph Gi¾?VMi¾?{v} is either K2n or Kn,n for some n for any vertex v of G and any minimal matching M such that {v} is a component of Gi¾?VM. We use this result to improve the upper bounds on the maximum number of vertices of 2-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus g to 4g+17 if gi¾?2 and to 12g+5 if gi¾?3. Moreover, for any nonnegative integer g we construct a 2-connected equimatchable factor-critical graph with genus g and more than 42g vertices, which establishes that the maximum size of such graphs is i¾?g. Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2-connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face-width k. Finally, we prove that any d-degenerate 2-connected equimatchable factor-critical graph has at most 4d+1 vertices, where a graph is d-degenerate if every its induced subgraph contains a vertex of degree at most d.

Efficient enumeration of rooted maps of a given orientable genus by number of faces and vertices

We simplify the recurrence satisfied by the polynomial part of the generating function that counts rooted maps of positive orientable genus g by number of vertices and faces. We have written an optimized program in C++ for computing this generating function and constructing tables of numbers of rooted maps, and we describe some of these optimizations here. Using this program we extended the enumeration of rooted maps of orientable genus g by number of vertices and faces to g = 4 , 5 and 6 and by number of edges to g = 5 and 6 and conjectured a further simplification of the generating function that counts rooted maps by number of edges. Our program is documented and available on request, allowing anyone with a sufficiently powerful computer to carry the calculations even further.

Determining Edge Expansion and Other Connectivity Measures of Graphs of Bounded Genus

In this paper, we give the first polynomial-time algorithm for determining the edge expansion for a graph of fixed orientable genus. We show that for a multigraph $G$ with $m$ edges and orientable genus $g$, the edge expansion of $G$ can be determined in time $m^{O(g)}$. We show that the same is true for various other similar measures of edge connectivity.

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.

Embedding 3-manifolds with boundary into closed 3-manifolds

We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere. This is a corollary of the following main result. Let M be a compact connected orientable 3-manifold with boundary. Denote G = Z , G = Z / p Z or G = Q . If H 1 ( M ; G ) ≅ G k and ∂ M is a surface of genus g, then the minimal group H 1 ( Q ; G ) for closed 3-manifolds Q containing M is isomorphic to G k − g . Another corollary is that for a graph L the minimal number rk H 1 ( Q ; Z ) for closed orientable 3-manifolds Q containing L × S 1 is twice the orientable genus of the graph.

Asymptotic Enumeration of Labelled Graphs by Genus

We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus $g$ with respect to vertices and edges simultaneously. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled $k$-connected graphs of orientable genus $g$ for $k\le3$.

The genus of Petersen powers

For each k, 1≤k≤n, we construct a dot product of n copies of the Petersen graph whose orientable genus is precisely k. We show that these are all possible values for the genus of Pn. This result gives counterexamples of all possible genera to a conjecture of Tinsley and Watkins from 1982. We show that the Petersen graph is the only Petersen power which can be embedded into the projective plane. For each k, 2≤k≤n − 1, we construct a Petersen power Pn whose non-orientable genus is precisely k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:1-8, 2011

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.

Local Rings with Genus Two Zero Divisor Graph

To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices are adjacent if their product is zero. Presently, we enumerate the local finite commutative rings whose zero divisor graphs have orientable genus 2.