The $\Delta$ property: a bridge between split graphs and Number Theory

For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $\Phi(S)$, a multigraph whose induced cycles have length at most $4$. In this paper we address the following question: for which $n \in \mathbb{N}$ is there a split graph $S$ whose factor graph contains an $n$-simple triangle, that is, a triangle all of whose edges have multiplicity $n$? We show that the answer is governed by a purely arithmetic condition, the $\Delta$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between Graph Theory and Number Theory.