Cycle-saturated graphs of minimum size
Given a graph \(F\), a graph \(G\) is said to be \(F\)-saturated if \(F \nsubseteq G\) but \(F \subseteq G + e\) for every edge \(e\) in the complement of \(G\). For \(n \geq |V(F) |\), we define \(\text{sat} (n,F)\) to be the smallest size of an \(F\)-saturated graph of order \(n\). The paper presents a series of results that establish bounds on \(\text{sat} (n, C_k)\), where \(C_k\) stands for a cycle of length \(k\). It is shown that with the exceptions of \(k = 8\) and \(k = 10\), we have \(n + c_1 n/k \leq \text{sat} (n, C_k) \leq n + c_2 n/k\). Several more specific results are also presented.
scientific article; zbMATH DE number 895985
scientific article