What is "panconnectivity"

In graph theory, a panconnected graph is an undirected graph in which, for every two vertices and , there exist paths from to of every possible length from the distance up to , where 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 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 has a Hamiltonian cycle, then the square of (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 is any connected graph, then the cube of (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 -vertex graph has degree at least , then the graph is panconnected.
• If an -vertex graph has at least edges, then the graph is panconnected.