Graphs with Many Hamiltonian Paths
Abstract
A graph is hamiltonian-connected if every pair of vertices can be connected by a hamiltonian path, and it is hamiltonian if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio of pairs of vertices connected by hamiltonian paths to all pairs of vertices approaches 1. We then consider minimal graphs that are hamiltonian-connected. It is known that any order-n graph that is hamiltonian-connected must have ≥ 3n/2 edges. We construct an infinite family of graphs realizing this minimum.
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