Counterexamples to the conjectured characterization of Hamiltonian degree sequences in digraphs

We show that the conjecture of Kühn, Osthus and Treglown (2008), which would characterize the degree sequences forcing a Hamilton cycle in a digraph, fails for every order $n\ge7$ and holds for $3\le n\le6$. It states that every digraph whose separately sorted out- and in-degree sequences satisfy two Chvátal-type conditions, and a third condition when the order is even, has a Hamilton cycle. It strengthens Nash-Williams' conjecture of 1975, which assumes strong connectivity instead of the third condition. Guninski found 6-vertex counterexamples to Nash-Williams' conjecture on MathOverflow in 2013 and 2020; the 2013 example was confirmed there by Royle and Thomas and the 2020 ones by DeBiasio. All of them violate the third condition. Our counterexamples are an explicit 7-vertex digraph and an explicit family, and the small orders are settled by exhaustive search. Hence Nash-Williams' conjecture fails for every $n\ge7$, in addition to Guninski's case $n=6$, and the same search shows that it holds for $3\le n\le5$. The family also shows that for every $n\ge8$ the conditions of either conjecture do not even force a cycle through every pair of vertices, answering a question recorded by Kühn and Osthus after Bermond and Thomassen. The counterexamples are checked in Lean. This work has not yet been reviewed by experts. The proofs, the Lean formalization and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.