Vol. 16, No. 1, 1966

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The enumeration of Hamiltonian polygons in triangular maps

Ronald C. Mullin

Vol. 16 (1966), No. 1, 139–145

A finite nonseparable topological graph G in E2 is said to be a triangular map if all its finite faces are triangular. Edges and vertices of G are external if they are incident with the infinite face, otherwise they are internal. The maps considered are rooted by distinguishing an external vertex and incident external edge. A polygon in the graph G of such a map is Hamiltonian if it includes afl vertices of G. In this paper, the average number of Hamiltonian polygons in a member of the class of nonisomorphic rooted triangular maps with n internal and m + 3 external vertices is determined. Asymptotic estimates are included for the results obtained. An unexplained coincidence is shown between the number of Hamiltonian polygons in rooted triangular maps and in their duals, rooted nonseparable trivalent maps.

Mathematical Subject Classification
Primary: 05.65
Received: 14 April 1964
Revised: 4 June 1964
Published: 1 January 1966
Ronald C. Mullin
University of Waterloo