On vertex-transitive graphs with a unique hamiltonian cycle
Combinatorics
2023-04-20 v2
Abstract
A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by A. Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.
Cite
@article{arxiv.2303.11124,
title = {On vertex-transitive graphs with a unique hamiltonian cycle},
author = {Babak Miraftab and Dave Witte Morris},
journal= {arXiv preprint arXiv:2303.11124},
year = {2023}
}