Minimizing the number of edges in [formula omitted]-saturated graphs

Abstract

Given a family of graphs F, a graph G is said to be F-saturated if G does not contain a copy of F as a subgraph for any F∈F, but the addition of any edge e∉E(G) creates at least one copy of some F∈F within G. The minimum size of an F-saturated graph on n vertices is called the saturation number, denoted by sat(n,F). Let C≥r be the family of cycles of length at least r. Ferrara et al. (2012) gave lower and upper bounds of sat(n,C≥r) and determined the exact values of sat(n,C≥r) for 3≤r≤5. In this paper, we determine the exact value of sat(n,C≥r) for r=6 and 28≤n2≤r≤n and give new upper and lower bounds for the other cases.