Infinite uniquely hamiltonian graphs

Importance: Medium ✭✭
Author(s): Mohar, Bojan
Recomm. for undergrads: no
Posted by: Robert Samal
on: July 24th, 2007

\begin{problem} Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $r > 2$? \end{problem}

(Originally appeared as [M].)

Let $G$ be a locally finite infinite graph and let $I(G)$ be the set of ends of~$G$. The \Def[Freudenthal compactification]{End_(topology)} of $G$ is the topological space $|G|$ which is obtained from the usual topological space of the graph, when viewed as a 1-dimensional cell complex, by adding all points of $I(G)$ and setting, for each end $t \in I(G)$, the basic set of neighborhoods of $t$ to consist of sets of the form $C(S, t) \cup I(S,t) \cup E'(S,t)$, where $S$ ranges over the finite subsets of $V(G)$, $C(S, t)$ is the component of $G - S$ containing all rays in $t$, the set $I(S,t)$ contains all ends in $I(G)$ having rays in $C(S, t)$, and $E'(S,t)$ is the union of half-edges $(z,y]$, one for every edge $xy$ joining $S$ and $C(S,t)$. We define a \emph{hamilton circle} in $|G|$ as a homeomorphic image $C$ of the unit circle $S^1$ into $|G|$ such that every vertex (and hence every end) of $G$ appears in $C$. More details about these notions can be found in [D].

A graph $G$ (finite or infinite) is said to be uniquely hamiltonian if it contains precisely one hamilton circle.

For finite graphs, the celebrated \OPrefnum[Sheehan's conjecture]{480} states that there are no $r$-regular uniquely hamiltonian graphs for $r>2$; this is known for all odd $r$ and even $r > 23$. For infinite graphs this is false even for odd $r$ (e.g. for the two-way infinite ladder), but each of the known counterexamples has at least 2 ends, leading to the problem stated.

Another way to extend Sheehan's conjecture to infinite graphs is to define \emph{degree of an end $t \in I(G)$} to be the maximal number of disjoint rays in $t$ and ask the following:

\begin{problem} Are there any uniquely hamiltonian locally finite graphs where every vertex and every end has the same degree $r > 2$? \end{problem}

Bibliography

[D] R. Diestel, Graph Theory, Third Edition, Springer, 2005.

*[M] Bojan Mohar, \href[Problem of the Month]{http://www.fmf.uni-lj.si/~mohar/Problems/P0703_HamiltonicityInfinite.html}


* indicates original appearance(s) of problem.