A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality

Abstract

We give a connected chordal graph on 15 vertices and 30 edges with Roman domination number 7 and double Roman domination number 10. It refutes the candidate inequality (3/2) gamma_R(G) <= gamma_dR(G) recorded at rank 67 of AutoGraphForge's public audit, together with its equivalent rank-68 restatement. The graph consists of a clique K5 and one independent vertex adjacent exactly to each pair of clique vertices. More generally, for the analogous split graph G_k, the manuscript proves gamma_R(G_k) = 2k - 3 for k >= 3 and gamma_dR(G_k) = 2k for k >= 4. Their ratio tends to one, so no universal multiplier greater than one lower-bounds gamma_dR/gamma_R on connected chordal graphs. The finite witness is checked on the original graph and original label alphabets. Author: Seth Christopher — Independent Researcher. Version 1.0.0. Files in this itemRoman_Double_Roman_Counterexample.pdf is the complete mathematical preprint. Roman_Double_Roman_Manuscript_Source_and_Evidence_v1.0.0.zip contains its LaTeX source, exact finite inputs, selected original research evidence, fresh public-source replay receipts, and a preservation map. This authored manuscript/evidence subset is offered under CC BY 4.0 to the extent applicable rights are held. Mathematical facts are not claimed as newly copyrighted; cited works retain their own rights. Companion verification softwareThe executable source is published separately under MIT as “Verification software for A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”, file Roman_Double_Roman_Verification_Software_v1.0.0.zip. Extract both ZIPs into one parent directory, merging their shared Roman_Double_Roman_v1.0.0 folder. The source-only replay requires Linux, Python 3.10+ and no third-party packages. This public release excludes private knowledge, embedded runtimes and unrelated investigations; it is not a redistribution of the full private agent archive. Verification and limitsFresh public-source replay completed 15 bounded serial processes. The standalone check accepted the exact 15-vertex certificate, verified its upper witnesses and chordality, and exhausted all 42,129 Roman assignments of weight below seven and all 1,076,624 double Roman assignments of weight below ten. The original differential suite passed 1,323,108 assertions across 1,098 labeled graphs on two through five vertices, with 24 relabelings and nine deliberately invalid or unsupported certificates rejected by both production checkers. Additional protocol and strict-JSON rejections passed. These checks share authorship and are not independent peer review. No global priority, minimum-order counterexample, or proof-assistant-kernel certification is claimed. AI assistance: ChatGPT provided substantial assistance with question selection, the graph construction and proofs, source research, verification software, test execution, and manuscript preparation. Aurelia performed bounded candidate search; Selene checked adjacency-set and reduced-label obligations; Mica checked the full original label alphabets and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.