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arXiv · 2510.11046

A recursive definition for the polymatroid Tutte polynomial

Xiaxia Guan·Xian'an Jin·Weiling Yang
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Abstract

The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently.

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Xiaxia Guan, Xian'an Jin, Weiling Yang. 2025-10-13. A recursive definition for the polymatroid Tutte polynomial. https://arxiv.org/abs/2510.11046

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