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. Williamson.^[1]

Each possible pair of vertices ${\displaystyle s}$ and ${\displaystyle t}$ have paths of length 1 through ${\displaystyle n-1}$, where ${\displaystyle n}$ is the number of vertices. Thus, the graph shown is panconnected.

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:

• 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.^[1]
• 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.^[1]
• If every vertex in an n-vertex graph has degree at least n/2 + 1, then the graph is panconnected.^[2]
• If an n-vertex graph has at least (n − 1)(n − 2)/2 + 3 edges, then the graph is panconnected.^[2]