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 − 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.

[1] 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 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:

Each possible pair of vertices and have paths of length 1 through , where is the number of vertices. Thus, the graph shown is panconnected.