# Plane Graph - Graph Theory - Homework, Lecture notes for Theory of Computation. Aligarh Muslim University

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Main points of this homework are: Plane Graph, N-Vertex Graph, Self-Dual Plane Graph, Vertices and Girth, Outerplanar Graph, 4-Color-Theorem, Use Network Flows, Spanning Trees, Complete Graph on Vertices, Directed Graph...
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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.