Jackson–Ordaz Hamiltonicity and pancyclicity conjectures

Jackson and Ordaz conjectured that a graph with connectivity exceeding its independence number is pancyclic; the conjecture is dated 1986 in the detailed source and 1990 in a survey. The Hamiltonicity analogue follows from the Chvátal–Erdős theorem.

Known results

• Chvátal–Erdős, 1972: $\kappa(G)\geq\alpha(G)$ implies Hamiltonicity.
• Keevash–Sudakov, 2009–2010: $\kappa(G)\geq600\alpha(G)$ implies pancyclicity; also $|V(G)|\geq150\alpha(G)^3$ suffices for Hamiltonian graphs.
• Amar, Fournier, and Germa: the conjecture holds for $\alpha(G)\in\{2,3\}$.
• Draganić, Munhá-Correia, and Sudakov: for every $\varepsilon>0$, sufficiently large graphs with $\kappa(G)\geq(1+\varepsilon)\alpha(G)$ are pancyclic.

2023 asymptotic threshold and 2026 directed-graph advance

Letzter’s 2023 paper reports that sufficiently large graphs with $\kappa(G)>\alpha(G)$ are pancyclic, an eventual exact-threshold result rather than a theorem for every graph. A 2026 preprint claims the related directed-graph bound $f_2(a)=O\!\left(a(\log a)^4/(\log\log a)^2\right)$ and a pancyclicity counterexample, but these claims lack independent assessment.

Current status (as of October 2026): Hamiltonicity is settled, pancyclicity is reported for sufficiently large graphs at the conjectured threshold, but the all-finite-graph statement and the related directed-graph linear bound remain open; the 2026 advance is unverified.