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Facet-defining inequalities of the symmetric Traveling Salesman Problem (TSP)polytope play a prominent role in both polyhedral TSP research andstate-of-the-art TSP solvers. In this paper, we introduce a new class offacet-defining inequalities, the \emph{circlet inequalities}. Theseinequalities were first conjectured in Gutekunst and Williamson \cite{Gut19b}when studying Circulant TSP, and they provide a bridge between polyhedral TSPresearch and number-theoretic investigations of Hamiltonian cycles stemmingfrom a conjecture due to Marco Buratti in 2017. The circlet inequalitiesexhibit circulant symmetry by placing the same weight on all edges of a givenlength; our main proof exploits this symmetry to prove the validity of thecirclet inequalities. We then show that the circlet inequalities arefacet-defining and compute their strength following Goemans \cite{Goe95b}; theyachieve the same worst-case strength as the similarly circulant crowninequalities of Naddef and Rinaldi \cite{Nad92}, but are generally stronger.

The Circlet Inequalities: A New, Circulant-Based Facet-Defining Inequality for the TSP