Neighborhoods and Covering Vertices by Cycles

G

=(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, \(\), and the minimum degree of the induced graph <X> is at least 3, then X is covered by one cycle.

This result will be in fact generalised by considering tuples instead of pairs of vertices.

Let \(\) be the minimum degree in the induced graph <X>. For any \(\),

\(\).

If \(\), and \(\), then X is covered by at most (p-1) cycles of G. If furthermore \(\), (p-1) cycles are sufficient.

So we deduce the following:

Let p and t (\(\)) be two integers.

Let G be a 2-connected graph of order n, of minimum degree at least t. If \(\), and \(\), then V is covered by at most \(\) cycles, where k is the connectivity of G.

If furthermore \(\), (p-1) cycles are sufficient.

In particular, if \(\) and \(\), then G is hamiltonian.