Planar Hypohamiltonian Graphs on 40 Vertices
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer‐aided generation of certain families of planar graphs with girth 4 and a fixed number of 4‐faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices.
Construction of planar triangulations with minimum degree 5
In this article, we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3-connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5.
The Use of Qualitative Information in Macro-Economic Policy Analysis
The majority of macro-economic policy models takes for granted that conflicts among objectives can be assessed by means of a tradeoff analysis. Such tradeoffs imply that sacrifices and benefits of various impacts of decisions can be measured on a cardinal scale, so that a conventional macro-economic utility or welfare analysis can be applied.