Discoveries / combinatorics

Dean's conjecture for $k=5$

Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for $k\in\{3,4\}$ and they proved every $k\ge6$, leaving $k=5$ open. If this proof stands, Dean's conjecture holds for all $k\ge3$. Nine finite proposition families in the bipartite and triangle-free branches are computer-assisted, with verifiers and certificates in a separately archived supplement. The reductions from arbitrary graphs to those finite state spaces are prose arguments in the paper and are not machine-checked. The claim has not been refereed, and no independent mathematician has audited the graph-theoretic core. The author describes extensive model-assisted hostile auditing of his own argument, which is worth something but is not external review, and says so plainly.