On Transition Polynomials of 4-Regular Graphs

Let G be a 4-regular graph. For every vertex v of G, there are three distinct possible ways of splitting v into two vertices of degree two, which we call transitions at v. A transition system of G is a familyp = (p(v), v ∈ V(G)), where p(v) is a transition at v. If we perform all vertex-splittings associated to the transitions of p, we obtain a family of disjoint cycles; we denote the number of these cycles by c(p). We also denote by P(G) the set of transition systems of G.

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