# Colouring of lines in a polygon to find relations between people at a party

There are $$7$$ people at a party. Each of them has 2 friends, 2 enemies and 2 strangers. If all relations are mutual (i.e. if A is friend of B, then B is friend of A), find a way for such an arrangement.
Note: this is not the original question, so it's ok if you can't find a solution (I'm not sure there is one myself).
My attempt was to take a heptagon and colour all of its diagonals and sides red (enemy), blue (stranger) and green (friend). I'm not sure how to proceed, though as the number of cases gets really big.
I'm sharing a Geogebra link to experiment with this... https://www.geogebra.org/m/n7pmwgvd

To achieve that, notice that each type of relation forms a cycle on the vertices, e.g., as everyone has exactly two enemies, you can start with one person, go to their enemy, then to the next enemy and so on without repeating anyone. The same applies to the other two relations as well.

• So you mean it's gonna be symmetric about each vertice? Mar 8, 2021 at 8:34
• I mean that each type of relation generates a cycle, that is a permutation which covers all the vertices, doesn't repeat anything and there is an edge between the last and the first vertex. Using only edges of one colour, you can go from any vertex through all the remaining vertices back to the start. It doesn't necessarily have to be symmetric when drawn. If you're familiar with graph theory, your question is equivalent to "How to decompose a complete graph K_7 into Hamiltonian cycles" Mar 8, 2021 at 9:06