Question 124
A subgraph of $K_n$ is chosen such that the number of its edges is a multiple of $3$ and the degree of each vertex is an even number. Prove that we can assign an integer weight to each triangle of $K_n$ such that for each edge of the chosen subgraph, the sum of the weights of the triangles that contain that edge equals $1$, and for each edge that is not in the subgraph, this sum is $0$.