Show that a planar map M = M(G) can be 2 colored iff every vertex of G has even degree.
[The map M(G) of a graph G is the collection of its faces, which are to be colored so that no adjacent faces have the same color.]
[hint: if every vertex of G has even degree then G is a union of edge-disjoint cycles. for another solution, apply induction on the number of edges, and delete the edges of a cycle forming the boundry of a face of M(G).]
Colored maps are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.