**Graph Theory: Coloring Plane Graphs II**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York

(Some of the questions below are easy to answer but not so easy to prove, and others are not even so easy to even answer!)

1. Under what conditions is it possible to vertex color a plane graph with exactly 2 colors?

2. Under what conditions is it possible to vertex color a plane graph with exactly 3 colors?

3. Under what conditions is it possible to face color a plane graph exactly 2 colors?

4. Under what conditions is it possible to face color a plane graph with exactly 3 colors?

Facts:

A. (Appel and Haken) Every plane graph can be vertex colored with 4 or fewer colors.

B. (Appel and Haken) Every plane graph can be face colored with 4 or fewer colors.

C. (Tait) A 3-valent plane graph G is 3-edge colorable if and only if the faces of G can be properly colored with 4 colors

D. (Vizing) Every graph can be edge colored with Æ or Æ + 1 colors, where Æ is the max valence (degree) of any vertex in the graph.

5. Can you find a 4-valent plane graph which requires 5 colors to color its edges?

6. Can you find a 5-valent plane graph which requires 5 colors to color its edges?