On Connected \(m-K_2-\)Residual Graphs

Abstract

P. Erdős, F. Harary, and M. Klawe studied the \(K_n\)-residual graph and came up with some conjectures and conclusions about the \(m-K_n\)-residual graph. For connected \(m-K_2\)-residual graphs, they constructed an \(m-K_2\)-residual graph of order \(3m+2\) and proposed that \(3m+2\) is the minimum order, which remained unproven. In this paper, using operation properties of sets and other methods, we prove that the minimum order of connected \(m-K_2\)-residual graphs is indeed \(3m+2\).