A factor graph is a graph where each node is numbered and if x is a factor of y, then x and y are connected. A graph is planar if it can be drawn on paper with no lines crossing.

1) Create a planar factor graph with nodes numbered from 2 through 23.

2a) [easy] Show that no planar factor graph is possible with nodes numbered 2 through 32.
2b) [hard] Show that no planar factor graph is possible with nodes numbered 2 through 24.

3) If the nodes are numbered 1 though n, find the largest planar factor graph and prove that n+1 is impossible.

Tip: A finite graph is planar if and only if it does not contain as a subgraph either the complete graph K₅ or the complete bipartite graph K_3,3.

Following the path of my previous post, 18 is impossible, so the highest value for the largest such graph is 17; the limit could be lower though.

1   6
2  12
3  18

Posted by Charlie on 2021-08-17 08:23:07