EECS 455: Applied Graph Theory

Homework 10

The next subject we will cover is plane graphs. The questions below cover Euler’s Formula and duals of

plane graphs. Also take a look at Kuratowski’s Theorem.

Problem 1: Prove Euler’s Formula using induction on the number of vertices and edges of G. (The Diestel

book gives an induction only on edges.)

Problem 2: Prove that every connected plane graph has a vertex with degree less than 6.

Problem 3: Prove that a set of edges in a connected plane graph G forms a spanning tree of G if and only

if the duals of the remaining edges form a spanning tree of G∗

.

Problem 4: Let G and G∗ be mutually dual plane graphs. Prove that if G is 2-connected then G∗

is

2-connected.