P_{m}-saturated graphs with minimum size

Abstract

By \(P_m\) we denote a path of order \(m\). A graph \(G\) is said to be \(P_m\)-saturated if \(G\) has no subgraph isomorphic to \(P_m\) and adding any new edge to \(G\) creates a \(P_m\) in \(G\). In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given \(m\) and \(n\) find the minimum size \(sat(n;P_m)\) of \(P_m\)-saturated graph and characterize the graphs of \(Sat(n;P_m)\) - the set of \(P_m\)-saturated graphs of minimum size. They have solved this problem for \(n\geq a_m\) where \(a_m=\begin{cases}3\cdot 2^{k-1}-2 &\quad\text{ if }\quad m=2k,\, k\gt 2\\ 2^{k+1}-2 &\quad\text{ if }\quad m=2k+1,\, k\geq 2\end{cases}\). We define \(b_m=\begin{cases}3\cdot 2^{k-2} &\quad\text{ if }\quad m=2k,\, k\geq 3\\ 3\cdot 2^{k-1}-1 &\quad\text{ if }\quad m=2k+1,\, k\geq 3\end{cases}\) and give \(sat(n;P_m)\) and \(Sat(n;P_m)\) for \(m\geq 6\) and \(b_m\leq n\leq a_m\).

Keywords

• graph
• saturated graph
• extremal graph