Pairs of Hamiltonian circuits in 5-connected planar graphs

Abstract

Settling a problem raised by B. Grünbaum, 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.