Graphs with Many Hamiltonian Paths
Abstract
A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{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- graph that is hamiltonian-connected must have edges. We construct an infinite family of graphs realizing this minimum.
Cite
@article{arxiv.2106.13372,
title = {Graphs with Many Hamiltonian Paths},
author = {Erik Carlson and Willem Fletcher and MurphyKate Montee and Chi Nguyen and Jarne Renders and Xingyi Zhang},
journal= {arXiv preprint arXiv:2106.13372},
year = {2025}
}
Comments
v3: substantial re-writing, including new author. To appear in Involve. v2: 12 pages, 6 figures. Substantial re-write including new results and removing results already proven by others. v1: 16 pages, 7 figures