Cycle-saturated graphs of minimum size Original Research Article

A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c1n/k and n + c2n/k for some positive constants c1 and C2. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.