Antichains in the cycle continuous order

Definitions and mathematical statement

For a finite undirected graph $G$, its binary cycle space $\mathcal Z(G)$ is the collection of edge subsets $F\subseteq E(G)$ for which every vertex has even degree in $(V(G),F)$. Addition is symmetric difference of subsets, making this a vector space over $\mathbb F_2$. Parallel edges are distinct coordinates; a loop contributes two to its vertex's degree. The empty subset is included, and a member may be a union of several cycles rather than one connected cycle.

Given finite graphs $G,H$, an edge map $f:E(G)\to E(H)$ is cycle-continuous if \[ f^{-1}(F)\in\mathcal Z(G) \qquad\text{for every }F\in\mathcal Z(H). \] No map of vertices, preservation of incidence, injectivity, or surjectivity is required. The inverse image is the ordinary set-theoretic inverse image of an edge subset; in particular every source edge mapping to an edge of $F$ is included.

The problem asks whether there is an infinite sequence of finite graphs $G_1,G_2,\ldots$, each with at least one edge, such that for every distinct $i,j$ no cycle-continuous map $E(G_i)\to E(G_j)$ exists. Since the condition is quantified over ordered pairs, it excludes maps in both directions. Such a sequence is an infinite antichain under cycle-continuous mapping. The graphs in the sequence are finite individually, with no uniform order bound required; one infinite graph is not an answer to this antichain question.

Short English statement

Is there an infinite family of finite graphs between which no cycle-continuous edge map exists in either direction?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3223 (OPG-323), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Antichains in the cycle continuous order. Original problem and discussion. http://www.openproblemgarden.org/op/antichains_in_the_cycle_continuous_order.