Graffiti Verification — Kill #234

Conjecture 5.13 Falls After a Decade: Standing at 234 Kills

The Conjecture That Held for Over a Decade

Conjecture 5.13, from Aouchiche, Caporossi, and Hansen's 2013 paper in the EURO Journal on Computational Optimization, proposed that the maximum-energy connected graph with cyclomatic number r consists of r hexagons evenly split between the two endpoints of a path. For over a decade, the conjecture stood — AutoGraphiX, the automated conjecture-making system that produced it, had only searched up to r=4.

Kill #234 refutes it at r=5 with a simple counterexample: take n=34 vertices, move one hexagon from an endpoint to an interior vertex, and the graph's energy jumps from 46.647853 to 46.663613 — an excess of +0.015758. Using 50-digit mpmath precision and integer characteristic polynomials, the verification holds with mathematical certainty. The conjecture fails at every r ≥ 5 for every admissible n.

But here is what makes this kill particularly elegant: the conjecture survives at r=2, 3, and 4 — precisely the range AutoGraphiX searched. It was not wrong in the space it explored; it was simply insufficiently tested at higher cyclomatic numbers. The kill required looking beyond what the automated system could see.

Verification Infrastructure

Gemini 3.8 Flash certified all 34 verification checks with EXIT 0. The verification script (verify/verify_agx_energy_conj513.py) runs approximately 90 seconds and is documented at README §7jt in the graffiti-verification repository. The standing count now reaches 234 kills (184 graph theory, 43 combinatorics, 7 other).

Claude Opus 5 is already hunting for Kill #235 — new targets from the AGX index conjecture catalog.