bibkey: gonzalezdeleonwachs2026weighted authors: Rafael S. Gonzalez D’Leon; Michelle L. Wachs year: 2026 title: “Weighted bond posets and a new chromatic symmetric function” doi: 10.48550/arXiv.2608.08692 url: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13 claim: “Conjecture 4.13(2): for a graph G on n vertices with k connected components and a spanning subgraph H with the same k components, (-1)^(n-k)(mu_G-mu_H) is real-rooted.” strata_touched:

• D5/S0/Certificates/GonzalezDLeonWachsWeightedBondSource
• D5/S0/Certificates/GonzalezDLeonWachsThreeVertexMobius
• D5/S0/Certificates/GonzalezDLeonWachsWeightedBondDifferenceRefutation license: citation-only triage: anchor

Weighted bond posets and a new chromatic symmetric function

Section 2 defines weighted connected-block partitions, their refinement order, the singleton bottom, and the weighted bond-poset Mobius polynomial. Propositions 2.1 and 2.2 supply the structural background, including the component product. Theorem 3.1 and formula (3.4) relate these polynomials to the paper’s later constructions.

Conjecture 4.13(2) is stated for an arbitrary graph G on n vertices with k connected components and a spanning subgraph H having the same k; it asserts real-rootedness of (-1)^(n-k)(mu_G-mu_H). It does not impose connectedness. The formal result targets only part (2) as written.

Verified locator

• DOI: 10.48550/arXiv.2608.08692
• URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13
• Locator: arXiv:2608.08692v1, Section 2, Theorem 3.1, formula (3.4), and Conjecture 4.13(2).
• Retrieved source hash: SHA-256 1781237e65346eab7d5119fcf67abb2d795c505d7e99a26e572df891a760773e.

The bounded literature checks recorded for issue 8501 found no full resolution. arXiv:2609.15784v1 develops nestohedron recurrences but did not provide a located resolution of Conjecture 4.13(2); its Mobius-polynomial follow-up was described as in preparation. This is bounded status evidence, not a worldwide priority claim.