An Affine Linear Solution for the 2-Face Colorable Gauss Code Problem in the Klein Bottle and a Quadratic System for Arbitrary Closed Surfaces
| dc.creator | Lins, Sostenes | |
| dc.creator | Oliveira-Lima, Emerson | |
| dc.creator | Silva, Valdenberg | |
| dc.date | 2003-01-02 | |
| dc.date | 2003-06-08 | |
| dc.date.accessioned | 2026-07-07T04:54:14Z | |
| dc.date.available | 2026-07-07T04:54:14Z | |
| dc.description | Let $\bar{P}$ be a sequence of length $2n$ in which each element of $\{1,2,...,n\}$ occurs twice. Let $P'$ be a closed curve in a closed surface $S$ having $n$ points of simple auto-intersections, inducing a 4-regular graph embedded in $S$ which is 2-face colorable. If the sequence of auto-intersections along $P'$ is given by $\bar{P}$, we say that is a {\em $P'$ 2-face colorable solution for the Gauss Code $\bar{P}$ on surface $S$} or a {\em lacet for $\bar{P}$ on $S$}. In this paper we present a necessary and sufficient condition yielding these solutions when $S$ is Klein bottle. The condition take the form of a system of $m$ linear equations in $2n$ variables over $\Z_2$, where $m \le n(n-1)/2$. Our solution generalize solutions for the projective plane and on the sphere. In a strong way, the Klein bottle is an extremal case admitting an affine linear solution: we show that the similar problem on the torus and on surfaces of higher connectivity are modelled by a quadratic system of equations. | |
| dc.description | 15 pages, 1 figure minor revisions relative to previous version | |
| dc.identifier | https://arxiv.org/abs/math/0301012 | |
| dc.identifier | http://arxiv.org/abs/math/0301012 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66167 | |
| dc.subject | Combinatorics | |
| dc.subject | primary 05C10; secondary 05C85 | |
| dc.title | An Affine Linear Solution for the 2-Face Colorable Gauss Code Problem in the Klein Bottle and a Quadratic System for Arbitrary Closed Surfaces | |
| dc.type | text |