Connectivity and Hamilton Circles in Locally Finite Graphs

Definitions and mathematical statement

Use the conventional setting of undirected simple graphs. A graph is locally finite if every vertex has finite degree. A ray is a one-way infinite simple path; two rays define the same end if no finite vertex set separates their tails. Write $\Omega(G)$ for the set of ends. Under the vertex-degree convention, define \[ d_G(\omega)=\sup\{|\mathcal R|:\mathcal R\text{ is a family of pairwise vertex-disjoint rays in }\omega\}. \] The source does not explicitly distinguish this convention from edge-degree of an end. A graph is $k$-connected if it has at least $k+1$ vertices and remains connected after deletion of fewer than $k$ vertices.

The record contains three distinct questions.

1. Does there exist $f:{\mathbb{Z}}_{\geq1}\to{\mathbb{Z}}_{\geq1}$ such that for every $k\geq1$ and every locally finite graph $G$, \[ \left(d_G(v)\geq f(k)\ \forall v\in V(G),\quad d_G(\omega)\geq f(k)\ \forall\omega\in\Omega(G)\right) \ \Longrightarrow\ \exists H\subseteq G\text{ that is }k\text{-connected}? \] The displayed convention permits $H$ to be finite or infinite; the source does not separately discuss a requirement that $H$ be finite or induced.
2. Does every $4$-connected planar locally finite graph have a Hamilton circle?
3. Does the square $G^2$ of every $2$-connected locally finite graph $G$ have a Hamilton circle? Here $G^2$ has vertex set $V(G)$ and an edge between distinct vertices at $G$-distance one or two.

For a connected locally finite graph, its Freudenthal compactification $|G|$ is the geometric graph with its ends adjoined, using the topology determined by components after finite vertex deletions. A Hamilton circle is a subspace homeomorphic to $S^1$ containing every vertex. In the third question the circle is in $|G^2|$. This topological notion, which accounts for ends, is not merely a two-way infinite Hamilton path.

The supplied literature triage checked 21 August 2026 reports the square-of-a-$2$-connected-graph assertion solved, the $4$-connected planar assertion only partially resolved, and the vertex/end-degree forcing question unsettled in general. These three status qualifications are retained separately and are not independent current-status findings.

Short English statement

The entry asks whether large degrees at both vertices and ends force a highly connected subgraph, whether every locally finite $4$-connected planar graph has a Hamilton circle, and whether every locally finite $2$-connected graph has a Hamilton circle in its square. The supplied assessment reports only the square assertion as fully solved.

Sources

• Local UnsolvedMath registry, permanent ID 30000022, exact source code catalogue entry; Connectivity and Hamilton Circles in Locally Finite Graphs. Proposed year 2004, status partially_solved; corrected_verified extraction retains all three subquestions.
• Combinatorics (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/01, pp. 5--110, https://doi.org/10.4171/owr/2004/01.
• Supplied literature assessment checked 2026-08-21: "Georgakopoulos", https://doi.org/10.1016/j.aim.2008.09.014; "Cui--Wang--Yu", https://yu.math.gatech.edu/Papers/circle20.pdf; "official report", https://ems.press/content/serial-article-files/45848. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims. No linked PDF was accessed here.