Timeline for Generalizations of the four-color theorem
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2014 at 20:46 | comment | added | Matt Samuel | The problem is only interesting for surfaces because every graph can be embedded in a manifold of dimension 3 or higher. | |
| Dec 9, 2014 at 20:14 | comment | added | DavidLHarden | $K_{7}$ in the torus has a nice algebraic description: Start with the graph formed by the Eisenstein integers, where $a$ is adjacent to $b$ means $|a-b| = 1$. Every vertex of this graph has degree 6, and it's planar. To make the plane a torus, we quotient by a lattice. To make this well-defined for how we identify vertices and edges, that lattice has to be an ideal. Choose an ideal of norm 7, like $(2+\sqrt{-3})$, and we now have 7 vertices (all have degree 6), yielding $K_{7}$. I don't know if replacing the Eisenstein ring with structures like the Hurwitz ring gives nice generalizations. | |
| Dec 8, 2014 at 8:51 | comment | added | william rood | Indeed so. How about if each country may consist of n disjoint regions? | |
| Dec 8, 2014 at 3:36 | comment | added | Todd Trimble | I guess that you mean that a (cartographic) map on a Klein bottle is colorable by 6 colors, etc. This is mentioned (and generalized) in the Wikipedia article: en.wikipedia.org/wiki/Four_color_theorem#Generalizations | |
| Dec 8, 2014 at 2:07 | review | Low quality posts | |||
| Dec 8, 2014 at 9:59 | |||||
| Dec 8, 2014 at 1:52 | review | First posts | |||
| Dec 8, 2014 at 5:59 | |||||
| S Dec 8, 2014 at 1:50 | history | answered | william rood | CC BY-SA 3.0 | |
| S Dec 8, 2014 at 1:50 | history | made wiki | Post Made Community Wiki by william rood |