Panconnectivity

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

In graph theory, a panconnected graph is an undirected graph in which, for every two vertices Template:Mvar and Template:Mvar, there exist paths from Template:Mvar to Template:Mvar of every possible length from the distance Template:Math up to Template:Math, where Template:Mvar 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 Template:Math 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 Template:Mvar has a Hamiltonian cycle, then the square of Template:Mvar (the graph on the same vertex set that has an edge between every two vertices whose distance in Template:Mvar is at most two) is panconnected.
• If Template:Mvar is any connected graph, then the cube of Template:Mvar (the graph on the same vertex set that has an edge between every two vertices whose distance in Template:Mvar is at most three) is panconnected.
• If every vertex in an Template:Mvar-vertex graph has degree at least Template:Math, then the graph is panconnected.
• If an Template:Mvar-vertex graph has at least Template:Math edges, then the graph is panconnected.

Related concepts

Vertex-pancyclic graphs: A graph of order Template:Mvar is vertex-pancyclic if every vertex lies on cycles of every possible length from the graph's girth up to Template:Mvar. While vertex-pancyclic graphs need not be panconnected, they share the property of having rich cycle structures.

Hamilton-connected graphs: These are graphs where every pair of vertices is connected by a Hamiltonian path. All panconnected graphs are Hamilton-connected, but the converse is not true. For example, the Template:Math graphs (line graphs of certain inclusion graphs) are Hamilton-connected for Template:Math but not panconnected.