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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