We denote by \(G(n)\) the graph obtained by removing a Hamilton cycle from the complete graph \(K_n\). In this paper, we calculate the lower bound for the minimum number of monochromatic triangles in any \(2\)-edge coloring of \(G(n)\) using the weight method. Also, by explicit constructions, we give an upper bound for the minimum number of monochromatic triangles in \(2\)-edge coloring of \(G(n)\) and the difference between our lower and upper bound is just two.