Chvátal’s toughness conjecture

Establish a constant t_0 such that every t_0-tough graph on at least three vertices is Hamiltonian.

Background

Toughness is a graph-connectivity parameter intended to capture necessary conditions for Hamiltonicity. Chvátal conjectured that there is a universal toughness threshold above which every graph with at least three vertices must contain a Hamilton cycle. The paper notes that non-Hamiltonian t-tough graphs exist for every t less than 9/4, so any valid threshold must be at least 9/4.

The paper proves the conjecture for 11-tough graphs that are (2P_2 \cup P_1)-free, and places this result within a broader program verifying the conjecture for particular graph classes. The general conjecture remains unresolved.

References

In 1973, Chvatal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. While this conjecture is still open, work has been done to confirm it for several graph classes, including all $F$-free graphs for every 5-vertex linear forest $F$ other than $P_5$ and $2P_2\cup P_1$.

— Hamilton cycles in tough $(2P_2 \cup P_1)$-free graphs  (2506.12684 - Shan et al., 15 Jun 2025) in Abstract; formally stated in the Introduction as Conjecture (Chvátal's toughness conjecture)

Despite these partial results, Conjecture 1.1 remains open.

— Hamiltonian cycles and Hamiltonian paths in $2k$-connected, $1$-tough and $(P_{3}\cup kP_{1})$-free graphs  (2608.13963 - Liu et al., 14 Aug 2026) in Section 1, Introduction