The Cycle Double Cover Conjecture is a famous unsolved problem in graph theory, which states that "every undirected graph without bridges has a collection of cycles that includes every edge exactly twice."
Simply put, the conjecture asserts that you can redraw all the edges of a graph twice and perfectly cover the original graph using only those cycles.
Origin: Proposed independently in the 1970s by mathematicians George Szekeres and Paul Seymour.
Background: Given any bridge-less graph, it asks whether you can create a set of disjoint cycles such that every edge is traversed exactly twice.
Current Status: Despite extensive research by numerous mathematicians, it remains an open problem, neither fully proven nor disproven. However, GPT-5.6 Sol Ultra has provided a proof.
The prompt is also publicly available. PDF 2 pages.
https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf
▶ Original source: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf
▶ Original source: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf
