bibkey: hausmann2026weightedmaxcut
authors: Will Hausmann
year: 2026
title: "Weighted Max Cut in Subcubic Graphs"
doi: null
url: https://arxiv.org/html/2611.05178v1
claim: "Question 5.4 asks for an independent side and residual component order at most three in every subcubic graph of odd girth exactly five."
strata_touched:
  - D5/S3/Combinatorics/Graph/ComponentPacking/HausmannQuestion54Refutation
license: CC-BY-4.0
triage: anchor

Independent-side partitions at odd girth five

Verified locator

• Primary HTML: https://arxiv.org/html/2611.05178v1
• Source scope: Section 1 definitions of finite simple graphs, subcubic degree and odd girth; Section 5, Question 5.4.

Hausmann, Weighted Max Cut in Subcubic Graphs, arXiv:2611.05178v1, Section 1 and Question 5.4 in Section 5, fixes finite simple graphs, subcubic degree at most three, and odd girth as the shortest odd cycle length. Question 5.4 reads:

Does every subcubic graph G of odd girth exactly five admit a partition V(G)=L∪R such that L is independent and every component of G[R] has order at most three?

The formal claim quantifies over every finite vertex presentation and simple graph, without a size cutoff. It retains both partition conditions, independence, and the order bound for every actual induced connected component. The odd-girth predicate requires a simple five-cycle and excludes every shorter odd simple cycle. Decidable finite presentations add no mathematical restriction.

The repository refutation uses the explicitly labelled GP(7,2). Its exact edge list and the deductive finite decision tree belong to the Lean source; the source paper is the supplier of the question, not of this refutation. The proof establishes a connected set of four residual vertices for every independent side. It asserts neither a size-minimal counterexample nor priority.

Lemma 4.7 supplies a residual-order-six statement. Theorem 2.2 is a weighted-cut mechanism under a component-union bipartiteness condition. Neither supplies an order-three partition. The source states that an affirmative answer to Question 5.4 would imply a weighted 4/5 bound in its odd-girth class; the refutation of that sufficient structural route does not refute that bound. Questions 5.3 and 5.5 are separate.

The inspected primary HTML has SHA-256 49bc3e301c8bea370cc9e9114b211a64ef529a70d268016c6e41d83254abe2d3. The quoted question and these locators establish source correspondence, not worldwide openness or a novelty claim.