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arXiv · 2607.20299

On (3,1)-regular graphs with one more vertex than edges

Frédéric Chyzak·Hui Huang·Manuel Kauers
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Abstract

Sequence A339987 of the OEIS counts (3,1)-regular graphs having one more vertex than edges by half the number of vertices. A recurrence relation satisfied by this sequence was guessed by Kauers and Koutschan in 2023. We confirm it in three ways: first, by a representation as the diagonal of a triple sum and an elaborate variant of traditional creative telescoping that makes an a posteriori validation possible; second, by a residue representation and a direct calculation by reduction-based creative telescoping; third, by a combinatorial recurrence on graph families and a calculation by differential elimination. Each of those three approaches leads to a formally complete proof and involves a computer calculation in one way or another.

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Frédéric Chyzak, Hui Huang, Manuel Kauers. 2026-07-22. On (3,1)-regular graphs with one more vertex than edges. https://arxiv.org/abs/2607.20299

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