Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number
Authors/Creators
- 1. Independent Researcher. Project association: Aletheia Technologies, an independent research project
Description
The manuscript proves an exact pendant-forest reduction, a parity property for rooted-tree responses, a singular attachment lemma in the graph-port setting, and a rank-one criterion determining when a pendant leaf increases the line-graph signature.
It establishes the universal upper bound s(L(G)) ≤ c(G). Together with the constructive lower bound, this gives floor((c+1)/2) ≤ f(c) ≤ c, so f(c) is finite and its maximum is attained for every cyclomatic number c. The sharper inequality 2s(L(G)) ≤ c(G)+1 remains a conjecture.
The load-bearing mathematical certificates and principal computational results were independently checked. The repaired R2 graph6 package corrects labeled graph6 serialization; the previous edge-list calculations remain valid.
Andrea Paone and Marco Paone contributed equally to this work.
Status: Preprint; not peer reviewed.
Notes
Version 1.3, revised on 30 July 2026. Just grammar forms
Version 1.2, revised on 29 July 2026, preserves the first-version date of 27 July 2026. It incorporates the universal upper bound s(L(G)) ≤ c(G), and hence floor((c+1)/2) ≤ f(c) ≤ c; updates obsolete statements about the finiteness of f(c); updates the targeted related-work discussion; and includes the repaired R2 graph6 certificates. The R2 repair corrects the labeled graph6 serialization; the previous edge-list calculations remain valid. The sharper inequality 2s(L(G)) ≤ c(G)+1 remains a conjecture.
Status: Preprint; not peer reviewed.
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line_graph_signature_beyond_2core_v1.3.pdf
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Additional details
Related works
- Is supplemented by
- Preprint: 10.5281/zenodo.21499790 (DOI)
- Preprint: 10.5281/zenodo.21534809 (DOI)