Orientable Surface Of Genus Research Articles

Automorphisms and isomorphisms of maps in linear time

A map is a \(2\) -cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no “truly subquadratic” algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus \(g\neq 0\) , parametrized by the genus \(g\) . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex \(v\) does not depend on the choice of \(v\) . The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.

Minimum Cuts in Surface Graphs

We describe algorithms to efficiently compute minimum $(s,t)$-cuts and global minimum cuts of undirected surface-embedded graphs. Given an edge-weighted undirected graph $G$ with $n$ vertices embedded on an orientable surface of genus $g$, our algorithms can solve either problem in $g^{O(g)} n \log \log n$ or $2^{O(g)} n \log n$ time, whichever is better. When $g$ is a constant, our $g^{O(g)} n \log \log n$ time algorithms match the best running times known for computing minimum cuts in planar graphs. Our algorithms for minimum cuts rely on reductions to the problem of finding a minimum-weight subgraph in a given $\mathbb{Z}_2$-homology class, and we give efficient algorithms for this latter problem as well. If $G$ is embedded on a surface with $b$ boundary components, these algorithms run in $(g + b)^{O(g + b)} n \log \log n$ and $2^{O(g + b)} n \log n$ time. We also prove that finding a minimum-weight subgraph homologous to a single input cycle is NP-hard, showing it is likely impossible to improve upon the exponential dependencies on $g$ for this latter problem.

The $$\mathbb {Z}_2$$-Genus of Kuratowski Minors

A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most two common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler $\mathbb{Z}_2$-genus of graphs.

Distance 4 curves on closed surfaces of arbitrary genus

Let Sg denote a closed, orientable surface of genus g≥2 and C(Sg) be the associated curve graph. Let curves α, β and the Dehn twist of α about β represent vertices a, b and c in C(Sg), respectively. If a and b are at a distance 3 from each other then we show that c is at a distance 4 from a and 3 from b. This produces many tractable examples of distance 4 vertices in C(Sg). As an application we show that the minimum intersection number of any two curves at a distance 4 on Sg is at most (2g−1)2.

Fractional Cocoloring of Graphs

The cochromatic number Z(G) of a graph G is the fewest number of colors needed to color the vertices of G so that each color class is a clique or an independent set. In a fractional cocoloring of G a non-negative weight is assigned to each clique and independent set so that for each vertex v, the sum of the weights of all cliques and independent sets containing v is at least one. The smallest total weight of such a fractional cocoloring of G is the fractional cochromatic number \(Z_f(G)\). In this paper we prove results for the fractional cochromatic number \(Z_f(G)\) that parallel results for Z(G) and the well studied fractional chromatic number \(\chi _f{(G)}\). For example \(Z_f(G)=\chi _f(G)\) when G is triangle-free, except when the only nontrivial component of G is a star. More generally, if G contains no k-clique, then \(Z_f(G)\le \chi _f(G)\le Z_f(G)+R(k,k)\), where R(k, k) is the minimum integer n such that every n-vertex graph has a k-clique or an independent set of size k. Moreover, every graph G with \(\chi _f(G)=m\) contains a subgraph H with \(Z_f(H)\ge (\frac{1}{4} - o(1))\frac{m}{\log _2 m}\). We also prove that the maximum value of \(Z_f(G)\) over all graphs G of order n is \(\varTheta (n/\log n)\), and the maximum over all graphs embedded on an orientable surface of genus g is \(\varTheta (\sqrt{g} / \log g)\).

On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold M3 of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus g, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than 12g. In this paper, we determine a class of diffeomorphisms GSDR(M3)⊂GSD(M3) that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class GSDR(M3) have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed g geq1, fixed number of periodic points, and any integer n≥12g, there exists a manifold M3 and a diffeomorphism f∈GSD(M3) having exactly n non-compact heteroclinic curves.

Intersections of Multicurves on Small Genus Non–Orientable Surfaces

Let $K_{n}$ ($n>1$) be an $n$--punctured non--orientable surface of genus $2$ with 1 boundary component. We give formulae for calculating the geometric intersection number of an arbitrary multicurve with a relaxed multicurve on $K_{n}$ given their generalized Dynnikov coordinates.

The Choice Number Versus the Chromatic Number for Graphs Embeddable on Orientable Surfaces

We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $\Theta(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.

Surface codes and color codes associated with non-orientable surfaces

Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of 8pi for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2pi . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

Edge‐maximal graphs on orientable and some nonorientable surfaces

AbstractWe study edge‐maximal, noncomplete graphs on surfaces that do not triangulate the surface. We prove that there is no such graph on the projective plane is the unique such graph on the Klein bottle and is the unique such graph on the torus . In contrast to this for each we construct an infinite family of such graphs on the orientable surface of genus , that are edges short of a triangulation.

Classification of periodic transformations of an orientable surface of genus two

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

Edge-preserving maps of the nonseparating curve graphs, curve graphs and rectangle preserving maps of the Hatcher–Thurston graphs

Let [Formula: see text] be a compact, connected, orientable surface of genus [Formula: see text] with [Formula: see text] boundary components with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the nonseparating curve graph, [Formula: see text] be the curve graph and [Formula: see text] be the Hatcher–Thurston graph of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is closed and [Formula: see text] is a rectangle preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We also prove that these homeomorphisms are unique up to isotopy when [Formula: see text].

Finite rigid subgraphs of pants graphs

Let $S_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We identify a finite rigid subgraph $X_{g,n}$ of the pants graph $\mathcal P (S_{g,n})$, that is, a subgraph with the property that any simplicial embedding of $X_{g,n}$ into any pants graph $\mathcal P (S_{g',n'})$ is induced by an embedding $S_{g,n}\to S_{g',n'}$. This extends results of the third author for the case of genus zero surfaces.

Linear Operators That Preserve Two Genera of a Graph

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.

Phase transitions in graphs on orientable surfaces

Let be the orientable surface of genus and denote by the class of all graphs on vertex set with edges embeddable on . We prove that the component structure of a graph chosen uniformly at random from features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős‐Rényi random graph chosen uniformly at random from all graphs with vertex set and edges. It takes place at , when the giant component emerges. The second phase transition occurs at , when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from and has only been observed for graphs on surfaces.

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $$\chi _s$$ of simplicial complexes naturally generalize the chromatic number $$\chi _1$$ of a graph. In any fixed dimension d, the s-chromatic number $$\chi _s$$ of d-complexes can become arbitrarily large for $$s\le \lceil d/2\rceil $$ (Bing in The geometric topology of 3-manifolds, Colloquium Publications, vol 40, American Mathematical Society, Providence, 1983; Heise et al. in Discrete Comput Geom 52:663–679, 2014). In contrast, $$\chi _{d+1}=1$$ , and only little is known on $$\chi _s$$ for $$\lceil d/2\rceil <s\le d$$ . A particular class of d-complexes are triangulations of d-manifolds. As a consequence of the Map Color Theorem for surfaces (Ringel in Map color theorem, Grundlehren der mathematischen Wissenschaften, vol 209, Springer, Berlin, 1974), the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $$\chi _2$$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $$\chi _2$$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $$f=(127,8001,5334)$$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction (Heise et al. 2014) along with embedding results (Bing 1983) can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of Heise et al. (2014), we obtain a rather small triangulation of the 3-dimensional sphere $$S^3$$ with face vector $$f=(167,1579,2824,1412)$$ and 2-chromatic number 5.

On Loops Intersecting at Most Once

We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like $$g^{k+1}$$, up to a factor of $$\log g$$. The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are LERF.

Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus g if and only if all of its subarrangements of size at most \(4g+4\) are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call an arrangement of graphs.

Linear Operators That Preserve the Genus of a Graph

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.