scholarly journals Distance-Restricted Matching Extension in Triangulations of the Torus and the Klein Bottle

10.37236/2952 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Robert E.L. Aldred ◽  
Jun Fujisawa
Keyword(s):  
Projective Plane ◽  
Perfect Matching ◽  
Klein Bottle ◽  
Even Order

A graph $G$ with at least $2m+2$ edges is said to be distance $d$ $m$-extendable if for any matching $M$ in $G$ with $m$ edges in which the edges lie pair-wise distance at least $d$, there exists a perfect matching in $G$ containing $M$. In a previous paper, Aldred and Plummer proved that every $5$-connected triangulation of the plane or the projective plane of even order is distance $5$ $m$-extendable for any $m$. In this paper we prove that the same conclusion holds for every triangulation of the torus or the Klein bottle.

scholarly journals Maximal arcs in projective planes of order 16 and related designs

2019 ◽  
Vol 19 (3) ◽  
pp. 345-351 ◽  
Author(s):  
Mustafa Gezek ◽  
Vladimir D. Tonchev ◽  
Tim Wagner
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Maximal Arcs ◽  
Even Order

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

Enumeration of unsensed r-regular maps on the projective plane and the Klein bottle

2021 ◽  
Vol 344 (11) ◽  
pp. 112528
Author(s):  
Evgeniy Krasko ◽  
Alexander Omelchenko
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Regular Maps

On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces

2006 ◽  
Vol 49 (4) ◽  
pp. 624-627
Author(s):  
Masakazu Teragaito
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Orientable Surface ◽  
Dehn Surgery ◽  
Orientable Surfaces

AbstractFor a non-trivial knot in the 3-sphere, only integral Dehn surgery can create a closed 3-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

scholarly journals REDUCING DEHN FILLINGS AND SMALL SURFACES

2005 ◽  
Vol 92 (1) ◽  
pp. 203-223 ◽  
Author(s):  
SANGYOP LEE ◽  
SEUNGSANG OH ◽  
MASAKAZU TERAGAITO
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
The Other ◽  
Möbius Band ◽  
Dehn Fillings ◽  
Orientable Surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

Crossing Numbers of Complete Graphs

2017 ◽  
Author(s):  
Noam D. Elkies
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Complete Graphs ◽  
Open Problems ◽  
Basic Properties ◽  
Crossing Numbers

This chapter examines crossing numbers. When a particular graph is drawn on a given surface, what is the smallest possible number of crossings among the edges? The chapter is organized as follows. Section 1 introduces crossing numbers; reviews the surfaces D, R 2, S, M, P, and T and some connections between them; and gives some basic properties of the crossing numbers, culminating with the existence of P Σ‎ for any surface Σ‎. Sections 2–4 treat crossing numbers on the sphere, projective plane, and torus in turn. Section 5 lists some open problems suggested by this analysis, on the same three surfaces and also on the Klein bottle and beyond. We relegate to an appendix the computation of the integrals that figure in the bounds on P P and P T.

MATCHING PROPERTIES IN DOUBLE DOMINATION EDGE CRITICAL GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 151-160 ◽  
Author(s):  
HAICHAO WANG ◽  
LIYING KANG
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Even Order ◽  
Double Domination

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) < γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

Sign in / Sign up

Export Citation Format

Share Document