Ore's theorem states that in a finite graph $G$ with $|V(G)|=n$, there is a Hamiltonian path, provided that the sums of the degrees of 2 distinct, non-adjacent vertices is $\geq n$.

For countable graphs, such a statement cannot hold: consider the disjoint union of two copies of $K_\omega$.

How about if we restrict ourselves to *connected* countable graphs? Is then the following statement true?

If $G$ is a connected countable graph such that for distinct, non-adjacent vertices $v,w$ we have $\text{max}\{\text{deg}(v), \text{deg}(w)\} = \aleph_0$, there is an countable Hamiltonian path (extending in two ways, or beginning at one vertex.)

twovertices if you take $\max$ anyway? $\endgroup$ – Alex Degtyarev Feb 20 '15 at 11:25