Iran TST
Source: Iranian TST 2019, third exam day 2, problem 5
June 24, 2019
combinatorics
Problem Statement
A sub-graph of a complete graph with vertices is chosen such that the number of its edges is a multiple of and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero.Proposed by Morteza Saghafian