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.
| Subjects: | Combinatorics (math.CO); Number Theory (math.NT) |
| Cite as: | arXiv:2605.25264 [math.CO] |
| (or arXiv:2605.25264v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25264 arXiv-issued DOI via DataCite (pending registration) |
From: Victor Nicolas Schvöllner [view email]
[v1]
Sun, 24 May 2026 21:34:09 UTC (46 KB)
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