TITLE: Disjoint paths between four vertices
QUESTION [6 upvotes]: Consider the following property of an undirected graph: For any four distinct vertices $a,b,c,d$, there is a path from $a$ to $b$ and a path from $c$ to $d$ such that the two paths do not share any vertex.
Is there a name for this property, or has it been studied? It looks related to vertex connectivity but not quite the same. Also it seems that any graph satisfying this property must have vertex connectivity at least $2$.

REPLY [8 votes]: The property that you are describing is called $2$-linked.  More generally, we say that a graph is $k$-linked if it has at least $2k$ vertices and for all distinct vertices $s_1, \dots s_k, t_1, \dots, t_k$ there are $k$-vertex disjoint paths connecting $s_i$ to $t_i$ for all $i \in [k]$.  Note that every $k$-linked graph is $k$-connected.  The converse is not true.  For example, a cycle is $2$-connected but is not $2$-linked.
However, it is a classic theorem that there is a function $f(k)$ such that every $f(k)$-connected graph is $k$-linked.  The current best bound for $f(k)$ is due to Thomas and Wollan, where they prove that every $10k$-connected graph is $k$-linked.  The paper is titled An improved linear edge bound for graph linkages, and is available  here.