On some minimal graphs of the torus
β Scribed by R. Bodendiek; K. Wagner
- Publisher
- John Wiley and Sons
- Year
- 1985
- Tongue
- English
- Weight
- 259 KB
- Volume
- 9
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
Let
be the set of finite, simple and nondirected graphs being not embeddable into the torus. Furthermore let >4 be a partial order-relation and M, (r) the minimal basis of I'. In this paper we determine three graphs of M, (r) being embeddable into the projective plane and containing the subgraph W.
π SIMILAR VOLUMES
We show that if \(G\) is a graph embedded on the torus \(S\) and each nonnullhomotopic closed curve on \(S\) intersects \(G\) at least \(r\) times, then \(G\) contains at least \(\left\lfloor\frac{3}{4} r\right\rfloor\) pairwise disjoint nonnullhomotopic circuits. The factor \(\frac{3}{4}\) is best
The bichromaticity of a bipartite graph B is defined as the maximum value of r + s for which B has the complete bipartite graph K,, as a homomorphic image We determine the bichromaticity of any bipartite cylinder graph C2,, x P, or torus graph CZn x C , , In the process, w e disprove a conjecture of
## Abstract We investigate the minimization problem of the minimum degree of minimal Ramsey graphs, initiated by Burr et al. We determine the corresponding graph parameter for numerous bipartite graphs, including biβregular bipartite graphs and forests. We also make initial progress for graphs of l
We give necessary and sufficient conditions for a directed graph embedded on the torus or the Klein bottle to contain pairwise disjoint circuits, each of a given orientation and homotopy, and in a given order. For the Klein bottle, the theorem is new. For the torus, the theorem was proved before by
## Abstract We write __H__βββ__G__ if every 2βcoloring of the edges of graph __H__ contains a monochromatic copy of graph __G__. A graph __H__ is __G__β__minimal__ if __H__βββ__G__, but for every proper subgraph __H__β² of __H__, __H__β²βββ__G__. We define __s__(__G__) to be the minimum __s__ such th