Shortness exponents for polytopes which are k-gonal modulo n (original) (raw)
1982, Journal of Combinatorial Theory, Series B
Barnette's famous conjecture states that the family, denoted here by .9;, of all those simple 3-dimensional polytopes whose faces are even sided contains only Hamiltonian members. Malkevitch [8] raised the question whether the family 9: of all simple polytopes having only pentagons, decagons, 15gons, etc., as faces contains only Hamiltonian members or not. In order to generalize the preceding problems we consider the class 9", of all (always 3-dimensional) polytopes which are k-gonal modulo n. We say that a polytope is k-gonal mod& n (k < n) if all vertices have the same valency and each of its faces is an m-gon with m = k (mod n). The few regular polytopes are all Hamiltonian. We shall see that, for many values of k and n, the polytopes which are k-gonal modulo n may well be non-Hamiltonian, even if they are assumed to be simple (every vertex is 3-valent). The class of all simple k-gonal (modulo n) polytopes (k < n) will be denoted by 9;. Griinbaum and Walther [7] introduced the following "measure" of how short a longest circuit can be, called the shortness exponent and defined for any family ST of graphs a(F) = ",",$f (log h(G))/log u(G), where v(G) is the number of all vertices of G and h(G) the maximal circuit length in G. We identify polytopes with the graphs of their vertices and edges. The case of those simple polytopes which are k-gonal modulo 3 was treated by Zaks [ 141 and Walther [ 121. They proved that ~(9;
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