On Multiplicity of Triangles in $2$-Edge Colouring of Graphs

Abstract

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.