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bleuprint
Posts:
56
From:
Belgium
Registered:
1/19/08
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Strange property of Heawood's vertex character in MPG
Posted:
Nov 2, 2009 11:16 AM
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Given a MPG (Maximal Planar Graph, all faces are triangles, also the infinite face) on v vertices. We can give a +1 or -1 number to the triangles. If we add MOD3 the triangle numbers of the triangles adjacent to a vertex we get a vertex number of 0, 1 or 2 for that vertex.
It's always possible to give triangle numbers so that all vertices have a vertex number of 0. It's Heawood's equivalent formulation of the 4 color theorem ( Heawood's vertex character on the dual of a cubic map).
But does there exist a proof or disproof for the following statement:
Given all the 2^(2v-4) combinations of triangle numbers, then any set of v-2 vertex numbers has all the 3^(v-2) different combinations of vertex numbers if the two missing vertices are adjacent?
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