On a certain representation of the chromatic polynomial

Abstract

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to deduce a number of corollaries, in particular, the following. A) The chromatic polynomial of a connected planar graph G can be uniquely determined from its combinatory dual graph G^* (although the graph G itself isn't, in general, determined uniquely by G^*). B) If a planar graph G is different from the full graph K_3 and has exactly one (up to renaming of colors) proper coloring of vertices in three colors, then the graph G^* dual to graph G is also vertex colorable in three colors.