Composite planar coverings of graphs

AbstractWe shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for some n⩾1 and show that every planar regular covering of a nonplanar graph is such a composite covering

Weighted zeta functions for quotients of regular coverings of graphs

AbstractLet G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs

Recursive Formulas for Beans Functions of Graphs

In this paper, we regard each edge of a connected graph $G$ as a line segment having a unit length, and focus on not only the vertices but also any point lying along such a line segment. So we can define the distance between two points on $G$ as the length of a shortest curve joining them along $G$. The beans function $B_G(x)$ of a connected graph $G$ is defined as the maximum number of points on $G$ such that any pair of points have distance at least $x\u3e0$. We shall show a recursive formula for $B_G(x)$ which enables us to determine the value of $B_G(x)$ for all $x \leq 1$ by evaluating it only for $1/2 \u3c x \leq 1$. As applications of this recursive formula, we shall propose an algorithm for computing $B_G(x)$ for a given value of $x\leq 1$, and determine the beans functions of the complete graphs $K_n$

Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4

We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2) triangulations with all vertices of degree at least 4. The method is based on a series of reversible operations, termed reductions, which lead to a minimal set of triangulations in each family. Throughout the process the triangulations remain within the corresponding family. Moreover, for the family (1) these operations reduce to the well-known edge contractions and removals of octahedra. The main results are proved by an exhaustive analysis of all possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189; MTM 2010-2044

Diagonal Flips of Triangulations on Closed Surfaces Preserving Specified Properties

AbstractConsider a class P of triangulations on a closed surfaceF2, closed under vertex splitting. We shall show that any two triangulations with the same and sufficiently large number of vertices which belong to P can be transformed into each other, up to homeomorphism, by a finite sequence of diagonal flips through P. Moreover, if P is closed under homeomorphism, then the condition “up to homeomorphism” can be replaced with “up to isotopy.

Graphs and projective plaines in 3

Proper homotopy equivalent compact P2-irreducible and sufficiently large 3-manifolds are homemorphic. The result is not known for irreducible 3-manifolds that contain 2-sided projective planes, even if one assumes the Poincaré conjecture. In this paper to such a 3-manifold M is associated a graph G(M) that specifies how a maximal system of mutually disjoint non-isotopic projective planes is embedded in M, and it is shown that G(M) is an invariant of the homotopy type of M. On the other hand it is shown that any given graph can be realized as G(M) for infinitely many irreducible and boundary irreducible M

Generating punctured surface triangulations with degree at least 4

As a sequel of a previous paper by the authors, we present here a generating theorem for the family of triangulations of an arbitrary punctured surface with vertex degree ≥ 4. The method is based on a series of reversible operations termed reductions which lead to a minimal set of triangulations in such a way that all intermediate triangulations throughout the reduction process remain within the family. Besides contractible edges and octahedra, the reduction operations act on two new configurations near the surface boundary named quasi-octahedra and N-components. It is also observed that another configuration called M-component remains unaltered under any sequence of reduction operations. We show that one gets rid of M-components by flipping appropriate edges

Negative Effect of Mathematics Education in High Schools

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