Hamiltonian paths in polyhexes: The use of branching subgraphs to assist diagnosis of graph traceability

Abstract

The traceability of some of the smaller polyhexes is examined. (A graph is said to be traceable or to have a Hamiltonian path if it has a path visiting every vertex just once.) Most polyhexes are traceable, and an attempt is made to develop some practical guidelines for finding those that are not. A subgraph consisting of the branching vertices of a polyhex, and of any edges which join pairs of such vertices, is a useful tool for this purpose. The “principal resonance structures” of such a graph suggest ways of finding simpler spannng subgraphs of the polyhex that will often make its traceability, or lack of it, more obvious.