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 $G$ be a finite undirected graph, possibly with parallel edges. A bridge is an edge whose removal increases the number of connected components. A circuit is a connected $2$-regular subgraph; in a multigraph, two parallel edges form a circuit, and a loop alone may be treated as a circuit. A circuit list is a multiset, so repetitions are permitted. For an orientation of $G$, orienting the traversal of a circuit distinguishes forward and backward use of each edge relative to the fixed orientation.

Statement recorded in the supplied source.

Does every bridgeless graph admit a circuit list in which every edge occurs exactly twice? The source also records two directed strengthenings: for every orientation of every bridgeless graph, (i) does a list of circuits with chosen traversal directions use every edge once forward and once backward; and (ii) does such a list use every edge twice forward and twice backward? [S2]

Short English statement

Can every bridgeless graph be covered exactly twice by cycles, with repetitions allowed? The same source asks for oriented double and fourfold covers with equal forward and backward counts.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3161 (OPG-137). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Cycle double cover conjecture, original node 137, statement and mathematical discussion. https://www.openproblemgarden.org/op/cycle_double_cover_conjecture.