Construct a Graph with 7 Vertex, 21 Edges on 2D Plane
On a 2D aircraft, just how to construct a chart with 7 vertexes and also 21 sides? I attempted numerous mix yet could not appear to attract that sort of contrived chart on a paper.
Yet my understanding is that it is possible. So any person can aid me with it?
Although the software program listed below does not permit one to look for planarity you could locate it valuable in seeing illustrations of charts (not commonly reeled in the aircraft) with a handful of vertices and also defined variety of sides :
My hunch is you desire $K_7$ made use of the torus ; see http://www.amotlpaa.org/math/k7torus.html
To address you examine regarding why $K_5$ is not planar : If a full chart is planar, after that for every single $K_3$ subgraph, either every vertex (that is not component of the $K_3$) is inside the $K_3$ or outside the $K_3$ (or else there is a going across side from outdoors to inside the $K_3$).
So, if we try to attract $K_5$ vertex - by - vertex, after that we first attract a triangular $K_3$. We can position the 4 - th vertex either inside or beyond the triangular. In either instance, the illustration gotten will certainly resemble :
Now, any place you placed the 5 - th vertex, you will certainly create some $K_3$ subgraph for which the various other 2 vertices are not both within and also not both outdoors.