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

2026-07-22

AI Summary

This work rigorously verifies the recurrence relation conjectured by Kauers and Koutschan in 2023 for OEIS sequence A339987, which enumerates (3,1)-regular graphs whose number of vertices exceeds the number of edges by one. We provide a complete proof through three independent approaches: diagonal triple sums combined with creative telescoping, residue representations integrated with reduced creative telescoping, and combinatorial recurrences over graph families augmented by differential elimination. This study presents the first multi-faceted formal verification of the recurrence, synergistically blending symbolic computation, combinatorial analysis, and complex analysis. Key steps are automated via computer algebra systems, demonstrating the power of interdisciplinary methodology and computational assistance in advancing the study of combinatorial sequences.

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.

Problem

• (3,1)-regular graphs
• recurrence relation
• OEIS sequence A339987
• vertex-edge difference
• graph enumeration

Innovation

• creative telescoping
• diagonal of rational functions
• residue representation
• combinatorial recurrence
• differential elimination