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arXiv · 2605.25264

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

Victor N. Schvöllner
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Abstract

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.

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Victor N. Schvöllner. 2026-05-24. The $\Delta$ property: a bridge between split graphs and Number Theory. https://arxiv.org/abs/2605.25264

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