Sum related to Goodman's Formula

Question

Suppose I have a (improper) 2-coloring of $E(K_n)$. Define the graph of all red edges to be $G$ and the graph of all blue edges to be $G'$. Then Goodman's formula says that the total number of triangles in $K_n$ is $$ \Delta = \frac{1}{2} \left(\sum_{v \in V(G)} \binom{d_G(v)}{2} + \sum_{v \in V(G')}\binom{d_{G'}(v)}{2} - \binom{n}{3}\right)$$ Given this, what would the sum $$ S = \sum_{v \in V(G)} \binom{d_G(v)}{2} + \sum_{v \in V(G')}\binom{d_{G'}(v)}{2}$$ represent?

Answer 1

You are essentially taking $2$ different kinds of triangles: The ones that have $2$ edges of one color and $1$ edge of a different color and the ones that have $3$ edges of the same color. Notice that the former are counted once because you have to be in the vertex(index of the sums) in which the $2$ colors collide. Notice that the later is counted $3$ times because each of the vertex is giving once the triangle. So, $$S=\text{# of triangles with at least two colors}+3\times \text{# of triangles with same color}.$$ The formula above works because you eliminate the first possibility and then divide by $2$ so you were counting the number of triangles where all edges had the same color.