Let X be a planar graph, and Y its dual planar graph. Is there a formula for the chromatic polynomial of Y in terms of the chromatic polynomial of X?
What if X is a cubic planar graph? I mean if each vertex has degree 3?
2025-01-12 23:57:04.1736726224
Chromatic polynomial of dual graphs
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Take a (convex) pentagon, and draw two chords from one of its vertices. The graph has chromatic polynomial $x(x-1)(x-2)^3$, and its dual, $A$, is a square with one diagonal.
Take a square with one diagonal, and add a vertex joined to the two ends of the diagonal. This graph also has chromatic polynomial $x(x-1)(x-2)^3$. Its dual, $B$, is just a square.
$A$ and $B$ have different chromatic polynomials, so you can't get the chromatic polynomial (or even the chromatic number!) of the dual from the chromatic polynomial of the graph.
My guess would be that there are such examples even if you restrict to cubic graphs.