W v cycles in plane graphs

A cycle in a plane graphG is called aW _v cycle if it has a connected (or empty) intersection with each face of the graph. We show that if the minimum degree (G)3 thenG has aW _v cycle and the lengthw(G) of a longestW _v cycle is bounded by the number,f(G), of faces ofG. The classW of graphsG withw(G)=f(G) is completely characterized by an characterized by an inductive construction from two graphs, namelyK ₄ and a face merging of two copies ofK ₄ on one hand, and in terms involving Halin graphs and face merging on the other hand. Longest cycles in members ofW are investigated. The shortness coefficient ofW is proved to be between one-half and three-quarters inclusively.     

Equivelar maps on the torus

We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3,6}, {6,3} or {4,4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with n vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of n. The asymptotic behaviour for n→∞ is also discussed, and an example is given for n such that the number of distinct equivelar triangulations of the torus with n vertices is larger than n itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type p²+pq+q² (or p²+q², resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.).     

Bisingular maps on the torus

A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.     

Polynomial vector fields on the torus

In this paper it is shown that the structurally stable polynomial vector fields on the torusT ², with singularities, are open and dense in the set of such vector fields. Many kinds of distinct dynamical phenomena are also presented by a list of examples including the Cherry flows. The above result works for analytical vector fields onT ² with the same proof.     

Generating even triangulations on the torus

A triangulation on a surface $\mathbf{F}$ is a fixed embedding of a loopless graph on $\mathbf{F}$ with each face bounded by a cycle of length three. A triangulation is even if each vertex has even degree. We define two reductions for even triangulations on surfaces, called the 4-contraction and the twin-contraction. In this paper, we first determine the complete list of minimal 3-connected even triangulations on the torus with respect to these two reductions. Secondly, allowing a vertex of degree 2 and replacing the twin-contraction with another reduction, called the 2-contraction, we establish the list for all minimal even triangulations on the torus. We also describe several applications of the lists for solving problems on even triangulations.     

Complex homogeneous splines on the torus

We construct functions M_α which are piecewise homogeneous polynomials on the (d+1)-dimensional torus U^d+1. These functions possess complete symmetry with respect to the independent variables. The symmetry and homogeneous relations for these functions are exploited to obtain a recurrence relation and explicit representations. Furthermore, we show that , where ω=e^12x/k, 0≤j_t≤k−1, are linearly independent. By restricting M_α to U^d, we obtain the complex analogue of polynomial box splines on a (d+1)-direction mesh on U^d, which is a multivariate analogue of B-splines on the circle studied by I.J. Schoenberg^[8].     

The short ruler on the torus

ABSTRACT

We can shorten any path that links two given points by applying short ruler transforms iteratively. In this article we take a closer look at a short ruler process on the torus. The torus is a compact Riemannian manifold and at least a subsequence of the process converges to a geodesic between the two points. However, on compact Riemann manifolds there might exist different limit geodesics (with the same length). On the torus, the geodesics with the same length are isolated and the limit geodesic is unique.     

Generalized Latin squares on the torus

The notion of a Latin square is generalized. The natural object on which to define this extension is the torus. A theorem is proved which shows that the existence of a Latin square implies the existence of a linear Latin square, a Latin square with special form. The theorems in the paper are used to provide alternate proofs of results due to Pólya and Chandra (in relation to a problem of Moser). The inability to extend the results to orthogonal Latin squares is noted.     

Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus,the projective plane and the Klein bottle. Sensed maps on the torus

The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root darts. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation — so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain above-mentioned classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms — so-called unsensed maps.     

On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Type 3 diminimal maps on the torus

A polyhedral map on the torus is diminimal if either shrinking or removing an edge yields a nonpolyhedral map. We show that all such maps on the torus fall into one of two classes, type 2 and type 3, and show that there are exactly two type 3 ones, which are given explicitly.     

Drawing 4-Pfaffian graphs on the torus

We say that a graph G is k-Pfaffian if the generating function of its perfect matchings can be expressed as a linear combination of Pfaffians of k matrices corresponding to orientations of G. We prove that 3-Pfaffian graphs are 1-Pfaffian, 5-Pfaffian graphs are 4-Pfaffian and that a graph is 4-Pfaffian if and only if it can be drawn on the torus (possibly with crossings) so that every perfect matching intersects itself an even number of times. We state conjectures and prove partial results for k>5. The author was supported in part by NSF under Grant No. DMS-0200595 and DMS-0701033.     

A note on geodesic foliations on the torus

We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and we give several new examples.      

Analytic Hardy spaces on the quantum torus

Analytic Hardy and BMO spaces on the quantum torus are introduced. Some basic properties of these spaces are presented. In particular, the associated H 1-BMO duality theorem is proved. Finally, we discuss some possible extensions of the obtained results.     

Enumeration of platonic maps on the torus

Certain maps (graph embeddings) on the torus are counted, namely those with all faces triangles, respectively quadrilaterals, resp. hexagons, and all vertices having the same degree (which then must be 6, 4 or 3, resp.). These are the toroidal analogues of the spherical maps corresponding to the five Platonic solids. Techniques from combinatorics and number theory are applied to obtain the results.