Coincidence Properties for Maps from the Torus to the Klein Bottle

The authors study the coincidence theory for pairs of maps from the Torus to the Klein bottle. Reidemeister classes and the Nielsen number are computed, and it is shown that any given pair of maps satisfies the Wecken property. The 1-parameter Wecken property is studied and a partial negative answer is derived. That is for all pairs of coincidence free maps a countable family of pairs of maps in the homotopy class is constructed such that no two members may be joined by a coincidence free homotopy.     

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic.     

Nonrevisiting paths on surfaces

The nonrevisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for surfaces, the nonrevisiting path conjecture is known to be true for polyhedral maps on the sphere, projective plane, torus, and a Klein bottle. Barnette has provided counterexamples on the orientable surface of genus 8 and nonorientable surface of genus 16. In this note the question is settled for all the remaining surface except the connected sum of three copies of the projective plane.     

THE TOPOLOGICAL AND DYNAMICAL PROPERTIES OF NONORIENTABLE SURFACES

Four topological and dynamical properties of nonorientable surfaces are proved.Thefirst is that for every continuous flow defined on any nonorientable closed surface,thereexist periodic or singular closed orbits.In the case of the projective plane,it confirms aconjecture of professor Ye Yian-qian in his lecture notes“dynamical systems on surfaces”.Secondly,the author gives an exact upper bound of the number of closed curves onnonorientable surfaces,which do not intersect each other and the complement of their sumis still connected.The third is concerned with the upper and lower bound of the number ofthe periodic or singular closed orbits with certain properties.The last is related to theconnectedness of the complement of a lifting curve on two-fold covering space.The firstproperty may be considered as a generalization of Kneser theorem from Klein bottle togeneral nonorientable surfaces and the second as a generalization of[4]Theorem 9.3.6from orientable surfaces to nonorientable surfaces.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

Enumeration of three kinds of rooted maps on the klein bottle

In this paper, $\tilde 2$ -essential — essential rooted maps on the Klein bottle are counted and an explicit expression with the size as a parameter is given. Further, the numbers of singular maps and the maps with one vertex on the Klein bottle are derived.     

克莱茵瓶上奇异地图的色数

A map is singular if each edge is on the same face on a surface (i.e., it has only one face on a surface). In this paper we present the chromatic enumeration for rooted singular maps on the Klein bottle.     

A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups

We consider finite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We show that whenever a group covers a Klein bottle weak solenoidal space it covers groups as well, moreover it covers the product of two solenoids. The converse is not true, we give an example of group which covers groups with any finite number of sheets, but does not cover any Klein bottle weak solenoidal space.     

Regular maps on non-orientable surfaces

It is well known that regular maps exist on the projective plane but not on the Klein bottle, nor the non-orientable surface of genus 3. In this paper several infinite families of regular maps are constructed to show that such maps exist on non-orientable surfaces of over 77 per cent of all possible genera.     

Klein瓶上的奇异地图

It is well known that singular maps (i. e. ,those have only one face on a surface)play a key role in the theory of up-embeddability of graphs. In this paper the number of rooted singular maps on the Klein bottle is studied. An explicit form of the enumerating function according to the root-valency and the size of the map is determined. Further ,an expression of the vertex partition function is also found.     

关于循环图的曲面嵌入

该文集中探讨循环图的曲面嵌入性质.决定了所有循环图的最小亏格(其中包括可定向亏格与不可定向亏格)和最大亏格.对于固定的整数l(≥3)和充分大的 自然数n,只有一种方式将4 -正则循环图C(n,l)嵌入到环面上使得其每一个面都是4 -边形.特别地,循环图$C(2l+2,l)$在加入若干条新边后可以同时将环面与Klein瓶进行三角剖分.     

Chromatic sums of singular maps on some surfaces

A map is singular if each edge is on the same face on a sruface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper povides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enum erating functions of such maps are also derived.     

Spanning trees on hypercubic lattices and nonorientable surfaces

We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice in d dimensions under free, periodic, and a combination of free and periodic boundary conditions. Results are also obtained for a simple quartic net embedded on two nonorientable surfaces, a Möbius strip and the Klein bottle. Our results are based on the use of a formula expressing the spanning tree generating function in terms of the eigenvalues of an associated tree matrix. An elementary derivation of this formula is given.     

Approximation of common fixed points of weakly compatible pairs using the Jungck iteration

We show that the Jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces. The existence of coincidence points are also discussed for those pair of maps. The results are generalizations of well known results of the convergence of Picard iterations for single self maps of Banach spaces. In particular, the results improve, generalize and extend the recent results of Berinde [V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213 (2009) 348-354] and answers the open question posed in the paper.     

Vertex-transitive graphs and vertex-transitive maps

We consider vertex-transitive graphs embeddable on a fixed surface. We prove that all but a finite number of them admit embeddings as vertex-transitive maps on surfaces of nonnegative Euler characteristic (sphere, projective plane, torus, or Klein bottle). It follows that with the exception of the cycles and a finite number of additional graphs, they are factor graphs of semiregular plane tilings. The results generalize previous work on the genus of minimal Cayley graphs by V. Proulx and T. W. Tucker and were obtained independently by C. Thomassen, with significant differences in the methods used. Our method is based on an excursion into the infinite. The local structure of our finite graphs is studied via a pointwise limit construction, and the infinite vertex-transitive graphs obtained as such limits are classified by their connectivity and the number of ends. In two appendices, we derive a combinatorial version of Hurwitz's Theorem, and classify the vertex-transitive maps on the Klein bottle.     

The Improvement of the Kneser Theorem and Its Applications

In this paper, first it is proved that on the Mobius strip M there is a unique periodic orbit of the continuous flow f which is the generator of the fundamental group \pi_1(M), where f is tangent (or transversal) to the boundary and has no fixed point on M. Then the results of the Kneser theorem are augmented. On the base of. these two results, the classification theorems for M and the Klein bottle are given, which are some more profound than those given in [1]. At last, applying the improved Kneser theorem to i some continuous flows on torus, the author gets the results that there exist periodic orbits the number of which is even (at least 2), and describes some qualitative behaviors of the orbits. Moreover, some simple applications to the general nonorientable 2-manifold, particularly to the,projective plane, are also mentioned.     

Note on the cochromatic number of several surfaces

The cochromatic number of a graph G, denoted by z(G), is the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces an empty or a complete subgraph of G. In an earlier work, the author considered the problem of determining z(S), the maximum cochromatic number among all graphs that embed in a surface S. The value of z(S) was found for the sphere, the Klein bottle, and for the nonorientable surface of genus 4. In this note, some recent results of Albertson and Hutchinson are used to determine the cochromatic numbers of the projective plane and the nonorientable surface of genus 3. These results lend further evidence to support the conjecture that z(S) is equal to the maximum n for which the graph G_n = K₁ U K₂ U … U K_n embeds in S.     

Coincidence index for fiber-preserving maps an approach to stable cohomotopy

A Coincidence index in any generalized (multiplicative) cohomology theory is defined for certain pairs of maps between euclidean neighborhood retracts over a metric space B.By taking an adequate geometric equivalence relation between two such coincidence situations, groups FIX^k (B) and FIX^k (B,A), for A closed in B, k an integer, can be defined. The purpose of this paper is to show that these groups constitute a generalized multiplicative cohomology theory. Moreover, we show that the index determines an isomorphism between this theory and stable cohomotopy.     

Klein瓶上的双奇异地图

一个地图的每条边如果不是环就是割边(即该边的两边是同一个面的边界)，则称之为双奇异地图，本文研究Klein瓶上带根双奇异地图的计数问题，得到了此类地图以边数、平面环数、手柄上本质环数和又帽上本质环数为参数的计数公式，并得到了部分计数显式。     

COMMON FIXED POINT THEOREMS FOR MULTI-VALUED MAPS

We establish some results on coincidence and common fixed points for a twopair of multi-valued and single-valued maps in complete metric spaces.Presented theorems generalize recent results of Gordji et...