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Simple 3-polytopal graphs with edges of only two types and shortness coefficients. (English) Zbl 0586.05027
Simple planar 3-connected trivalent graphs are considered with only two types of faces: 5-gons and q-gons, in which no two q-gons have a common edge. For any \(q\geq 28\) infinitely many non-Hamltonian graphs are constructed. It also follows that for large enough q values any cycle in these graphs misses at least 1/60 of the vertices, thereby radically improving previous results. For all \(q\geq 16\) infinite families of graphs (not necessarily non-Hamiltonian) are constructed. It is also shown that that any graph with \(q\geq 6\) is cyclically 4-edge connected.
Reviewer: F.Plastria

MSC:
05C38 Paths and cycles
05C40 Connectivity
51M20 Polyhedra and polytopes; regular figures, division of spaces
05C45 Eulerian and Hamiltonian graphs
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