Odd cycles and low oddness

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.

A finite graph is cubic if every vertex has degree three, and bridgeless if deleting any one edge does not increase its number of connected components. A $2$-factor is a spanning subgraph in which every vertex has degree two, hence a union of disjoint cycles. For a bridgeless cubic graph define \[\omega(G)=\min\{\text{number of odd-length cycles in }F:F\text{ is a }2\text{-factor of }G\}.\]

Statement recorded in the supplied source.

If every cycle in every $2$-factor of a bridgeless cubic graph $G$ has odd length, must $\omega(G)\leq2$? [S2]

Short English statement

If every spanning cycle cover of a bridgeless cubic graph consists entirely of odd cycles, must some such cover have at most two cycles?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3109 (OPG-37182). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden contributors, Odd cycles and low oddness, Open Problem Garden, node 37182. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/odd_cycles_and_low_oddness.