Fractional Hamiltonicity of 2-tough graphs

Establish whether every 2-tough graph on at least three vertices admits a fractional Hamiltonian cycle.

Background

Fractional Hamiltonicity is a linear relaxation of Hamiltonicity in which edge weights in [0,1] have total weight equal to the number of vertices and assign weight at least two to every nontrivial edge cut. Scheinerman and Ullman conjectured that 2-toughness is sufficient for fractional Hamiltonicity. The paper proves the weaker universal bound that every 10-tough graph on at least three vertices is fractionally Hamiltonian, so the conjectured threshold 2 remains unresolved.

References

They conjectured that every $2$-tough graph is fractionally HamiltonianConjecture~2.3.6\footnote{Wang claimed to prove this conjecture. However, the proof as written does not establish the required upper bound on the dual optimum: the displayed dual-feasible solution of value $n$ shows only that $\opt(D)\ge n$, whereas one must prove $\opt(D)\le n$.}.

— Toughness Bounds for Fractional Hamiltonicity and Resistance Positivity  (2609.03412 - Wang, 3 Sep 2026) in Section 1, Introduction, paragraph beginning “Scheinerman and Ullman defined fractional Hamiltonicity”