Search in the document preview
HW 7: due Thursday, November 1 in class
1. A plane graph is called self-dual if it is isomoprhic to its dual.
(a) Show that if an n-vertex graph G is self dual, then G has 2n− 2 edges.
(b) For each n ≥ 4, construct a simple n-vertex self-dual plane graph.
2. Let G be a plane graph. Show that G∗∗ is isomorphic to G if and only if G is connected.
3. Show that if G is a connected planar graph with n vertices and girth k ≥ 3, then G has at most k(n− 2)/(k − 2) edges.
4. Show that if G is a simple planar graph with ≥ 11 vertices, then the complement of G is non-planar.
5. An outerplanar graph is a graph that can be embedded in the plane so that every vertex appears on the outside face. Use the 4-color-theorem to prove that any outerplanar graph is 3-colorable.
6. (Extra credit) Prove that every outerplanar graph is 3-colorable without using the 4-color theorem.