The problem
In every bridgeless (2-edge-connected) graph, there exists a collection of cycles such that every edge belongs to exactly two of them. The strong form asks whether eight cycles always suffice.
The Cycle Double Cover Conjecture
AI-resolvedReferences
History & significance
Emerged in the 1970s from work of Szekeres, and Seymour's 1979 formulation; connected to the Four Color Theorem and snark theory via Tutte. Special cases fell steadily (planar graphs, squares-free graphs, Petersen-minor-free graphs) but fifty years produced no general proof.
The resolution (solved by AI)
Proved — OpenAI's GPT-5.6 Sol, released 10 July 2026: every bridgeless graph admits a cycle double cover with at most eight cycles. Published alongside the prompt: hours-long effort budgets, parallel sub-agents cross-checking each other's work, and instructions not to give up — a methodology as notable as the mathematics. Human reviewers confirmed the argument combines previously known techniques with unusual patience rather than fundamentally new tools. As with the unit-distance disproof two months earlier, the pattern repeats: the machinery existed; no human had assembled it.