Title:

Hamiltonian circuits in trivalent planar graphs

The author has investigated the properties of Hamiltonian circuits in a class of trivalent planar graphs and he has attempted, with partial success, to establish conditions for the existence of Hamiltonian circuits in such graphs. Because the Hamiltonian circuits of a trivalent planar graph are related to the fourcolourings of the graph some aspects of the four colour problem are discussed. The author describes a colouring algorithm which extends the early work of Kempe, together with an algorithm based on the Heawood congruences which enables the parity of the number of fourcolourings to be determined without necessarily generating all of the fourcolourings. It is shown that the number of Hamiltonian circuits has the same parity as the number of fourcolourings and that the number of Hamiltonian circuits which pass through any edge of a trivalent planar graph is either even or zero. A proof is given that the latter number is nonzero, for every edge of the graph, whenever the family of fourcolourings has either of two stated properties. The author describes two original algorithms, independent of fourcolourings, which generate a family of Hamiltonian circuits in a trivalent planar graph. One algorithm embodies a transformation procedure which enables a family of Hamiltonian circuits to be generated from a given Hamiltonian circuit, while the other generates directly all Hamiltonian circuits which include a chosen edge of the graph. In a new theorem the author proves the existence of Hamiltonian circuits in any trivalent planar graph whose property is that one or more members of a family of related graphs has odd parity.
