Therefore, at least four colors are needed to color this graph and the plane containing it.

The procedure must be adjusted slightly because graphs might not contain any leaves.

That is, a 1-tree is a connected graph containing exactly one cycle.

The graph that is used is directed, and does not contain any cycles.

So this new graph does not contain any (r + 1)-clique.

Again, the new graph does not contain any (r + 1)-clique.

It is also true that every such geometric graph contains disjoint edges of the same color.

Every graph contains at most 3 maximal independent sets, but many graphs have far fewer.

A graph with nodes can contain at most bridges, since adding additional edges must create a cycle.

If the graph contains loops, then there may be multiple paths between the chosen nodes.