The Cycle Double Cover Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G=(V,E)$ be a finite undirected graph; parallel edges and loops are allowed. The degree of a loop at its vertex is counted twice. A cycle means a connected subgraph in which every vertex has degree two, so a loop and a pair of parallel edges are included among the possible cycles. An edge is a bridge if deleting it increases the number of connected components of $G$; $G$ is bridgeless if it has no such edge. A cycle double cover is a finite list of cycles $(C_1,\ldots,C_m)$, with repetitions permitted, such that every edge belongs to exactly two entries of the list. [S2]

Statement recorded in the supplied source.

Does every finite bridgeless undirected graph $G$ have cycles $C_1,\ldots,C_m$ satisfying \[ \forall e\in E(G),\qquad \sum_{i=1}^m\mathbf 1_{\{e\in E(C_i)\}}=2? \] Here the indicator $\mathbf 1_A$ is $1$ if the condition $A$ holds and $0$ otherwise. For an edgeless graph the empty list is allowed. No connectedness or cubic-degree hypothesis is imposed on $G$. [S2]

Short English statement

Can every graph with no bridge have its edges covered by a list of cycles so that each edge appears exactly twice?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 59 (GT-004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Sang-il Oum, A Proof of the Cycle Double Cover Conjecture by OpenAI: An Exposition, arXiv:2607.16356v3 (2026), Section 1, definitions and Theorem 1; Section 3 on loops and repeated cycles. https://arxiv.org/pdf/2607.16356.