Five cycle double cover conjecture

Status open high confidence

The five cycle double cover conjecture (Preissmann 1981, Celmins 1984) — that every 2-edge-connected graph has a double cover by at most five even subgraphs — remains open as of September 2026. It is strictly stronger than the (also open) cycle double cover conjecture and strictly weaker than the strong 5-cycle double cover conjecture. The only verified post-2008 paper directly addressing 5-cycle double covers provides a partial characterization (for cubic graphs) rather than a proof of the full statement.

Cited literature (1)

• partial A note on 5-cycle double covers (2012)
  Hoffmann-Ostenhof, Arthur · arXiv preprint · arXiv:1209.0096

  Establishes a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of a bridgeless cubic graph; does not resolve the full conjecture.

Reviewer notes. Web search was unavailable for this review; all evidence was gathered via direct URL fetches. The conjecture sits strictly between two other open problems tracked in the corpus: it is stronger than OPG:cycle_double_cover_conjecture (no bound on the number of even subgraphs) and weaker than OPG:strong_5_cycle_double_cover_conjecture (which additionally prescribes a specific circuit in the cover). The OPG:petersen_coloring_conjecture implies this conjecture but is itself open. The Open Problem Garden lists both the CDC and the strong 5-CDC as open with high importance; no primary source resolving the 5-CDC was found. A Wikipedia article contained a suspicious claim (consistent with model hallucination) that the CDC was proved by an AI model in July 2026; this was disregarded as unverified. The 2012 Hoffmann-Ostenhof paper is the only directly relevant post-2008 primary source retrieved; it proves a partial characterization for cubic graphs only.

Auto-reviewed 2026-09-10 with claude-sonnet-4-6 (web search enabled).