Forcing a 2-regular minor

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.

All graphs are finite and simple. The average degree of $G$ is $2|E(G)|/|V(G)|$. A graph $H$ is a minor of $G$ if it can be obtained by deleting vertices or edges and contracting edges. A $2$-regular graph has degree two at every vertex, hence is a vertex-disjoint union of cycles.

Statement recorded in the supplied source.

For every integer $t\ge3$, does every graph of average degree at least $4t/3-2$ contain every $2$-regular graph on $t$ vertices as a minor? [S2]

Short English statement

Does the proposed sharp average-degree threshold force every prescribed disjoint union of cycles as a minor?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3187 (OPG-59911). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Forcing a 2-regular minor, original node 59911, statement and mathematical discussion. https://www.openproblemgarden.org/op/forcing_a_2_regular_minor.