What is the cycle double cover conjecture?

The cycle double cover conjecture is a long-standing math problem in graph theory, originally posed in the 1970s, which asserts that every bridgeless graph has a collection of cycles covering each edge exactly twice. It can also be formulated in terms of graph embeddings as the circular embedding conjecture. In July 2026, a preprint claimed a positive resolution using a proof generated by OpenAI's GPT-5.6.

What we know

• A cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. [2, 3]
• The cycle double cover conjecture is the unproven assertion that every bridgeless graph has a cycle double cover. [2, 3]
• The problem was posed in the 1970s by W. T. Tutte, Itai and Rodeh, George Szekeres, and Paul Seymour. [2]
• In July 2026, OpenAI released a preprint claiming a positive resolution of the conjecture, with a proof generated by GPT-5.6. [2] · medium confidence

Where to go next

• the bigger pictureWhat is a graph embedding?
• whoWho was W. T. Tutte?
• a case in pointWhat are snarks in graph theory?
• how it workedHow do large language models generate mathematical proofs?
• compared withWhat is the circular embedding conjecture?
• an unexpected connectionHow are football strategies connected to graph theory?Both defensive football tactics and topological graph structures share the identical terminology of a double cover.

Sources

• [1]cycle double cover (Wikidata Q5198159) · CC0 1.0
• [2]Cycle double cover · CC BY-SA 4.0
• [3]Double cover · CC BY-SA 4.0
• [4]Snark (graph theory) · CC BY-SA 4.0
• [5]List of mathematical discoveries made with artificial intelligence · CC BY-SA 4.0