Neighborhoods and covering vertices by cycles

In this work, we show the following results. First if G=(V,E) is a 2-connec ted graph, and X is a set of vertices of G such that for every pair x,x' in X, \N-G(x)boolean OR N-G(x')\greater than or equal to n/2+2, and the minim um 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 pa irs of vertices. Let delta(1)(X) be the minimum degree in the induced graph <X>. For any t g reater than or equal to 2, delta(t)(X) = min{\N-G(u(1))boolean OR N-G(u(2))...boolean OR N-G(u(t))\,u( i)not equal u(j),u(1),...,u(t)is an element of X}. If delta(1)(X)greater than or equal to t, and delta(t)(X)greater than or eq ual to\V\/p+t, then X is covered by at most (p-1) cycles of G. If furthermo re delta(1)(X)greater than or equal to 2t, (p-1) cycles are sufficient. So we deduce the following: Let p and t (t greater than or equal to 2) be two integers. Let G be a 2-connected graph of order n, of minimum degree at least t. If d elta(1)greater than or equal to t, and delta(t)greater than or equal to n/p +t, then V is covered by at most (p-1) + inverted right perpendicular (t-1) /k inverted left perpendicular cycles, where k is the connectivity of G. If furthermore delta(1)greater than or equal to 2t, (p-1) cycles are suffic ient. In particular, if delta(1)greater than or equal to 2t and delta(t)greater t han or equal to n/2 + t, then G is hamiltonian.