Cyclically 5-edge-connected cubic planar graphs and shortness coefficients
It is shown that some classes of cyclically 5-edge-connected cubic planar graphs with only one type of face besides pentagons contain non-Hamiltonian members and have shortness coefficients less than unity.
Non-hamiltonian polyhedral graphs with two types of faces
Abstract In the class of cubic cyclically 5-connected polyhedral graphs with only 5-gons and q-gons ( q ⩾22) as faces there are non-hamiltonian members.
Pairs of Hamiltonian circuits in 5-connected planar graphs
Abstract Settling a problem raised by B. Grunbaum, J. Malkevitch, and the author, we present 5-valent 5-connected planar graphs that admit no pairs of edgedisjoint Hamiltonian circuits; our smallest example has 176 vertices. This is used to construct an infinite family of 5-valent 5-connected planar graphs, in which every member has the property that any pair of Hamiltonian circuits in it share at least about 1 168 of their edges. We construct 4- and 5-valent, 3-connected non-Hamiltonian planar graphs.