Panconnectivity

In graph theory, a panconnected graph is an undirected graph in which, for every two vertices $s$ and $t$, there exist paths from $s$ to $t$ of every possible length from the distance $d(s,t)$ up to $n &minus; 1$, where $n$ is the number of vertices in the graph. The concept of panconnectivity was introduced in 1975 by Yousef Alavi and James E. Williamson.

Panconnected graphs are necessarily pancyclic: if $uv$ is an edge, then it belongs to a cycle of every possible length, and therefore the graph contains a cycle of every possible length. Panconnected graphs and are also a generalization of Hamiltonian-connected graphs (graphs that have a Hamiltonian path connecting every pair of vertices).

Several classes of graphs are known to be panconnected:
 * If $G$ has a Hamiltonian cycle, then the square of $G$ (the graph on the same vertex set that has an edge between every two vertices whose distance in G is at most two) is panconnected.
 * If $G$ is any connected graph, then the cube of $G$ (the graph on the same vertex set that has an edge between every two vertices whose distance in G is at most three) is panconnected.
 * If every vertex in an $n$-vertex graph has degree at least $n/2 + 1$, then the graph is panconnected.
 * If an $n$-vertex graph has at least $(n &minus; 1)(n &minus; 2)/2 + 3$ edges, then the graph is panconnected.