Zero at interior positions on two 95-edge arms over a rooted graceful graph
Given any finite graph with a conventional graceful labeling that assigns zero to a chosen root, attach two 95-edge arms at that root. Each interior vertex on either new arm can receive zero in its own graceful labeling.
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Beedbyte
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Completed
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1
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cea70e05d2c95787dac0de09d9a726d5d7c05553
Let H be any finite graph with Q edges, a specified root r, and a conventional graceful labeling g: its vertex labels are distinct values in 0,…,Q, its edge differences are exactly 1,…,Q, and g(r)=0. Attach two separately named paths of 95 edges at r, identifying r with their common midpoint and retaining every vertex and edge of H. For either new arm and each depth d=1,…,94, the resulting graph has a conventional graceful labeling that assigns zero to the selected new vertex. The labeling may vary with the arm and depth; its labels are distinct in 0,…,Q+190 and its edge differences are exactly 1,…,Q+190.
The construction selects a certified 191-label path for each depth, shifts its high-side labels by Q, and attaches it to H by identifying the path midpoint with r and assigning each old vertex v the label 94+g(v). Existing alpha-amalgamation and label-complement operations support this splice; the result does not claim a new primitive operation. Earlier sources for the general alpha-vertex amalgamation include Huang–Kotzig–Rosa as stated by Panpa, Imnang and Wasuanankul (2025, Theorem 2.5) and by Shan and Zhong (2026, Lemma 1). Barrientos (2020) describes related transformations and a narrower two-alpha-input amalgamation.
The result is conditional on the supplied root-zero labeling of H. It covers the 94 interior depths on each new arm, with a separate labeling for each target. It does not assert zero-labelings at the new tips or at old vertices of H. The proof evidence includes a separate internal mathematical audit, which disclosed limited exposure to the author prose, and a separate internal copied-source Lean 4.34.0 replay of the exact theorem and its stated limits.
For any finite graph H with Q edges, any specified root r, and any conventional graceful labeling g:V(H)→{0,…,Q} with g(r)=0, form a graph by attaching two new paths of 95 edges at r and identifying r with their common midpoint. For each of the two new arms and every depth d=1,…,94, the resulting graph has a conventional graceful labeling assigning zero to that selected new vertex. The labeling may depend on the selected arm and depth.
Formal verification
Formally verified within the stated scope
Review status
Internal check
Evidence
The exact theorem has a fresh copied-source Lean 4.34.0 replay across 59 project modules, plus a separate mathematical audit and finite graph checks. Checks are internal to the project, not external review.
Scope
For any supplied conventional graceful root-zero labeling of a finite graph H, the graft retains every old H vertex and edge and adds two separately named 95-edge arms. Each new arm at depths 1..94 receives zero in its own target-dependent labeling. No claim for new depth-95 tips or old H target vertices. The output uses distinct vertex labels in 0..Q+190 and edge weights 1..Q+190; the labeling need not be onto.
Assumptions
A conventional graceful labeling is injective into 0..Q and its edge absolute differences are exactly 1..Q; the chosen root is supplied with label zero. The output is not asserted onto unless the supplied residual labeling is onto.
Internal project checks; no external review claimed.
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Contributions & AI assistance
Contributions
Jordi Gartner publishes this work through Beedbyte.
AI assistance
AI tools assisted the documented construction and proof-code work, path and graph checks, the copied-source replay, and preparation of the translation drafts.
Beedbyte
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