VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Every finite connected simple graph G satisfies
$$\alpha(G)\ge r(G)+\ln(\rho(G)),$$
where $\alpha(G)$ is the independence number, $r(G)$ is the radius, and $\rho(G)$ is the minimum number of pairwise vertex-disjoint paths whose vertices cover $V(G)$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Thinking
GPT-5.6 Thinking produced and checked an infinite family of counterexamples. For each integer s >= 0, it considered a tree T_s formed from the path v_0v_1...v_{4s+7} by attaching leaves at v_{2s+2} and v_{2s+5}. It proved that
$$\alpha(T_s)=2s+5,\qquad r(T_s)=2s+4,\qquad \rho(T_s)=3.$$
Since $\ln 3>1$, it follows that
$$\alpha(T_s)=2s+5<2s+4+\ln 3=r(T_s)+\ln\rho(T_s).$$
Thus every T_s is a counterexample, disproving the conjecture and providing infinitely many counterexamples. The AI also audited the final proof line by line.
The proof was checked line by line by GPT-5.6 Thinking. The radius, independence number, perfect matching, and path-covering number arguments were separately recomputed, including the smallest case s=0. The proof appears mathematically valid, but as of 2026-07-30 it has not been independently verified by an external graph theorist, a formal proof assistant, or peer review.