The $Δ$ property: a bridge between split graphs and Number Theory
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914598836240384 |
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| author | Schvöllner, Victor N. |
| author_facet | Schvöllner, Victor N. |
| contents | For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $Φ(S)$, 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 $Δ$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between Graph Theory and Number Theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $Δ$ property: a bridge between split graphs and Number Theory Schvöllner, Victor N. Combinatorics Number Theory For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $Φ(S)$, 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 $Δ$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between Graph Theory and Number Theory. |
| title | The $Δ$ property: a bridge between split graphs and Number Theory |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2605.25264 |