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Triangle on six vertices
Task number: 4206
Let G be a graph on six vertices. Prove that it contains either a triangle or an induced independent set on three vertices.
Solution
Choose any vertex v. By analyzing the cases, we verify that at least one of the following possibilities occurs:
- The vertex v has a degree of at least three and some of its neighbors are connected by an edge. Then it contains a triangle.
- The vertex v has a degree of at least three and none of its neighbors are connected by an edge. Then these neighbors form an independent set of size at least three.
- The vertex v has a degree of no more than two, and some of its non-neighbors are not connected by an edge. It then contains an independent set of size three.
- The vertex v has a degree of no more than two, and all of its non-neighbors are connected by an edge. Then these non-neighbors form a complete graph on at least three vertices and a triangle within.
