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On a class of Hamiltonian polytopes. (English) Zbl 0655.05045
Let $${\mathcal S}(p,q)$$ denote the class of simple 3-polytopal graphs all of whose edges are incident with two p-gons or a p-gon and a q-gon, $$p\neq q$$, p,q$$\geq 3$$. In this paper it is shown that all graphs of the class $${\mathcal S}(5,q)$$ for $$3\leq q\leq 12$$, $$q\neq 5$$ are Hamiltonian, using the following proposition due to S. K. Stein: A simple 3-polytopal graph G is Hamiltonian if and only if its dual $$G^*$$ has point- arboricity equal to 2 [Bull. Am. Math. Soc. 76, 805-806 (1970; Zbl 0194.560)]. The theorem in the present paper supplements the known results of P. R. Goodey that all graphs of the classes $${\mathcal G}(3,6)$$ and $${\mathcal G}(4,6)$$ are Hamiltonian, where $${\mathcal G}(p,q)$$ denotes the class of 3-connected 3-valent planar graphs, i.e. simple 3-polytopal graphs, all of whose faces are p-gons and q-gons, $$p<q$$, $$p\geq 3$$ [Isr. J. Math. 22, 52-56 (1975; Zbl 0317.05114); J. Graph Theory 1, 181-185 (1977; Zbl 0379.05037)].
Reviewer: I.Tomescu

##### MSC:
 05C45 Eulerian and Hamiltonian graphs
Full Text:
##### References:
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