Panconnectivity

Graph families

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. 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 − 1)(n − 2)/2 + 3 edges, then the graph is panconnected. (Wikipedia).

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Panconnectivity

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