Multivariate independence-polynomial and related partition-function bounds
Follow the latest progress by email.
How small can a graph’s multivariate independence polynomial be at nonnegative vertex weights, degree by degree? The related target bounds the partition function for semiproper colourings with two proper colours.
Why it matters and what counts as progress
Why it matters: It extends lower-bound questions from uniform activities to heterogeneous vertex weights and a broader colouring model. Equality characterization is beyond the restricted formal target.
What counts as progress: Sharper weighted graph-polynomial bounds or an extension to additional semiproper colouring models. An equality classification goes beyond the restricted formal target.
How the problem developed
- 12 Jun 2018 · Lower bound [1]
Ashwin Sah, Mehtaab Sawhney, David Stoner and Yufei Zhao Proved a univariate lower bound for the independence polynomial of a graph, providing an earlier benchmark for later multivariate extensions.
- 2 Feb 2026 · Generalization [2]
Joonkyung Lee and Jaehyeon Seo; a custom research agent powered by Gemini Deep Think generated some mathematical content Proved multivariate lower bounds extending the univariate result and related bounds for semiproper colourings and partition functions; a Lean formalization of Theorem 1.4 is linked by the authors.
AI contribution
Lower bounds for multivariate independence polynomials and their generalisations
Lee and Seo prove a degreewise lower bound for the multivariate independence polynomial at nonnegative vertex weights and a two-proper-colour semiproper partition-function bound. The model-origin Lemma 2.1 and dual-set ideas are retained central steps; the equality characterization in their first theorem is outside the Lean target.
Sources
Website timeline
Publication dates and editorial changes.