Unbounded Burning Number at Slater Number Two

Abstract

We construct a connected graph on 15 vertices with burning number 4 and Slater number 2, refuting the candidate inequality b(G) <= s(G) + 1 recorded in AutoGraphForge's public conjecture audit. The graph consists of a clique K7 joined by one edge to a path on eight new vertices. Its Slater number is determined by its two largest degrees. A ten-vertex geodesic excludes three burning rounds, while an explicit four-round schedule burns the whole graph. More generally, joining a path on t new vertices to K_(t-1), for t >= 8, gives s(G) = 2 and b(G) >= ceil(sqrt(t + 2)). Burning number is therefore unbounded on connected graphs with Slater number two. The proof uses the established diameter lower bound for graph burning. Complete finite certificates and reproducible verification accompany the self-contained arguments. Author: Seth Christopher — Independent Researcher. Scientific version 1.0.0. Files in this itemUnbounded_Burning_Number_at_Slater_Number_Two.pdf is the complete seven-page mathematical preprint. Burning_Slater_Manuscript_Source_and_Evidence_v1.0.0.zip contains the LaTeX source, finite certificate, tables, original research evidence, historical receipts, and fresh source-replay results. These materials are offered under CC BY 4.0, to the extent the licensor holds applicable rights. Mathematical facts and raw numbers are not claimed as newly copyrighted, and cited works retain their own rights. Companion verification softwareThe executable Python code is supplied separately under the MIT License in the companion Software item titled “Verification software for Unbounded Burning Number at Slater Number Two”, file Burning_Slater_Verification_Software_v1.0.0.zip. This separation preserves the paper/evidence and software licensing scopes. Extract both ZIPs into the same parent directory and merge their shared Burning_Slater_v1.0.0 directory. The software is offline and standard-library-only; the full source replay is supported on Linux with Python 3.10+. Verification and limitsThe finite checker evaluates all 3375 ordered three-center sequences for the 15-vertex graph, checks the capacity obstruction, and verifies the four-round witness. The full replay includes bounded search, checking, admission, regression tests and three finite-family instances. The infinite-family statement is established by the manuscript's mathematical argument, not by testing finitely many examples. These computational checks share authorship; they do not constitute independent peer review or a proof-assistant formalization. This work targets the burning–Slater candidate inequality, not the distinct square-root graph-burning conjecture. No claim of global priority or of a smallest counterexample is made. AI assistance: ChatGPT (OpenAI, used in this project as TPM45) provided substantial assistance with problem selection, mathematical development and proof checking, literature searches, verification-software preparation and execution, and manuscript drafting. Aurelia performed bounded candidate search; Selene checked the exhaustive three-round obstruction; Mica checked the capacity certificate and four-round witness. These are computational tools, not coauthors or independent reviewers. Seth Christopher initiated and directed the project and is the sole author responsible for the submitted content.