On the saturation spectrum of families of cycle subdivisions.

Given a family of graphs F, a graph G is F-saturated if G contains no member of F as a subgraph, but G+e contains a member of F as a subgraph for each edge e in the complement of G. The edge spectrum of F is the set of all possible sizes of F-saturated graphs on n vertices. A G-subdivision is a graph derived from G by replacing each edge of G with a path of arbitrary length. Let C≥k denote the family of Ck-subdivisions, where Ck is a cycle of length k with k≥3. Determining the minimum or maximum number of edges in n-vertex F-saturated graphs are two of the most important problems in the study of extremal graph theory. This is also a very important optimization problem in graph theory. The study of this problem is closely related to the development of other branches of mathematics, computer science, network, modern information science and technology. In this paper, we determine the edge spectrum of C≥k for each k∈{3,4,5,6}.