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Pentagonal 3-polytopal graphs with edges of only two types and shortness parameters

✍ Scribed by Stanislav Jendrol'; Peter J. Owens

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 510 KB
Volume: 137
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(93)e0141-p

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✦ Synopsis

We consider the class of pentagonal 3-polytopal graphs all of whose edges are incident either with two 3-valent vertices or with a 3-valent vertex and a q-valent vertex. For most values of q, (i) we find a small non-hamiltonian graph in the class and (ii) we show that the shortness exponent of the class and the shortness coefficient of a special subclass are less than one. For q--4, we find a positive lower bound for the shortness coefficient.

Maximum cycles in k-valent or k-gonal 3-polytopal graphs have been studied in [2-15] and elsewhere.

An edge of a planar graph is of type (i,j; p, q) if its end vertices have valencies i and j and it is incident with a p-gon and a q-gon. Apart from the five platonic graphs and four other graphs, every 3-connected planar graph has edges of at least two types.

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