Neighborhoods and covering vertices by cycles (Q5928568)

Let \(G\) be a graph on \(n\) vertices, \(\delta_t\) the minimal size of the union of the neighborhood of \(t\) vertices. If \(G\) is 2-connected with minimal degree 3, \(\delta_2\geq n/p+2\), then \(G\) can be covered by at most \(p-1\) cycles. If \(G\) is \(k\)-connected, has minimal degree \(\geq t\), \(\delta_t\geq n/p+t\) for some \(t\geq 3\), then \(G\) can be covered by \((p-1)+\lceil (t-1)/k\rceil\) cycles. If \(\delta_t\geq n/p+t\) and the minimal degree is greater than or equal to \(2t\), then \(G\) can be covered by at most \(p-1\) cycles.