Definition talk:Connectivity

What happens with this definition when applied to the full graph on a countable infinity of points? Then $\kappa$ cannot be finite, but the definition doesn't cover this at the moment. --Lord_Farin 03:32, 17 February 2012 (EST)

That's a good question. I don't have experience in infinite graph theory. Certainly one could generalize $k$-connectedness and hence connectivity to all the cardinals (and probably by modifying the definition of $k$-connectivity such that if $k$ is an infinite cardinal, then we must have that $|V(G)| \geq k$, where it is $|V(G)| > k$ for finite $k$). Off-hand, I think a number of important theorems such as Menger's Theorem ($k$-connectivity if and only if there are $k$ disjoint paths between any two vertices) would still hold. I assume that someone's already done this, but I haven't heard of it.