A Theorem on Indifference Graphs - Hilmer, Kieran; Pinchasi, Rom; Ponomarenko, Vadim (The PUMP Journal of Undergraduate Research, 2021)
An indifference graph P has as vertices a set of points on the real line, with an edge between every two points at most 1 apart. For every point x in P, we set N(x) to be the set of points that are adjacent to x. Fix an indifference graph P, and a positive integer k. Suppose that for every x in P, k divides |N(x)|. We show that the number of points in P is also divisible by k.