Abstract
Let G(V ,E ) be a connected plane graph with vertex–set V , edge-set E , set of regions R, and let G∗(R,E ) be its dual graph. The 4–Color Theorem (4–CT for short) states that the regions of G can always be colored with 4 colors such that adjacent regions receive different colors [3,16]. It is well–known that it suffices to verify the 4–CT for connected 3–regular bridgeless plane graphs, and that in this case the number of 3–edge colorings equals the number of 4–region colorings with the outer region colored 0 (see e.g. [2]). Thus the number of 3–edge colorings is an important parameter to study. In [15] Penrose derived a number of remarkable expressions for this parameter, defining implicitly what is now called the Penrose polynomial P (G , λ) in the variable λ. Among other things he showed for a 3–regular connected plane graph:
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