The cycle double cover conjecture states that every bridgeless graph has a collection of cycles which together contain every edge exactly twice.
This conjecture was independently formulated by Szekeres (1973) and Seymour (1979).
In July 2026, OpenAI released a claimed proof described as AI-generated (OpenAI 2026a, Pegg 2026), the prompt it reported led to the proof (OpenAI 2026b), and an accompanying Lean formalization for finite loopless bridgeless multigraphs (OpenAI 2026c). As of July 2026, the claim had not yet appeared in a refereed publication.
A dual form of the problem is called the Fulkerson
conjecture .
See also Bridgeless Graph ,
Cycle Double Cover ,
Fulkerson Conjecture ,
Hamiltonian Graph
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References Archdeacon, D. "The Cycle Double Cover Conjecture." http://www.cems.uvm.edu/~archdeac/problems/cyclecov.htm . Goddyn,
L. "Cycle Covers of Graphs." Ph.D. thesis. Waterloo, Ontario, Canada: University
of Waterloo. Jaeger, F. "A Survey of the Cycle Double Cover Conjecture."
In Cycles in Graphs (Ed. B. Alspach and C. D. Alspach). North
Holland, pp. 1-12, 1985. OpenAI. "A Proof of the Cycle Double
Cover Conjecture." July 9, 2026a. https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf . OpenAI.
"Prompt Used for 'A Proof of the Cycle Double Cover Conjecture'." July
9, 2026b. https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf . OpenAI.
"CDC Lean Formalization." GitHub repository, July 9, 2026c. https://github.com/openai/cdc-lean . Pegg,
E. Jr. "Get Snarky: The Cycle Double Cover Conjecture--An AI Proof?"
Wolfram Community , July 11, 2026. https://community.wolfram.com/groups/-/m/t/3756327 . Seymour,
P. D. "Sums of Circuits." In Graph Theory and Related Topics
(Ed. J. A. Bondy and U. R. S. Murty). New York: Academic
Press, pp. 341-355, 1979. Szekeres, G. "Polyhedral Decompositions
of Cubic Graphs." Bull. Austral. Math. Soc. 8 , 367-387, 1973. West,
D. "Cycle Double Cover Conjecture (1978/1979)." http://www.math.uiuc.edu/~west/openp/cdc.html . Referenced
on Wolfram|Alpha Cycle Double Cover Conjecture
Cite this as:
Weisstein, Eric W. "Cycle Double Cover Conjecture."
From MathWorld https://mathworld.wolfram.com/CycleDoubleCoverConjecture.html
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