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Graphs : Coloring Maps

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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).]

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Proof : Let G we any graph of 2 face colourable. Then dual graph of G., i.e. G* having chromatic number ...

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