Prescribed zero positions on two arms of a rooted graceful graft
For any finite graph H with a conventional graceful labeling and a chosen root labeled 0, attach two named arms of K=23+24t edges, where t≥1. On each new arm, a separate conventional graceful labeling can place zero at depth 20+22t or 21+22t.
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a8125ba821be4b89867d379501a81ebf95445cf2
Let H be any finite indexed graph with Q edges and a conventional graceful labeling g: its vertex labels are distinct elements of 0,…,Q, and its edge differences are exactly 1,…,Q. Choose an actual root r with g(r)=0. Attach two separately named paths of K=23+24t edges at r, identifying each path's initial vertex with r and retaining every vertex and edge of H. For each new arm, and separately for each depth d₀=20+22t and d₁=21+22t measured from r, the graft has a conventional graceful labeling that assigns zero to that selected vertex. These are four separate existence statements; the labeling may depend on the arm and depth. No tree, connectedness, alpha-labeling, or onto-vertex assumption is required of H. If g is not onto 0,…,Q, the output is still conventionally graceful but need not use every label in its full vertex range. No zero is prescribed at an old vertex of H, a new tip, or another depth.
The construction uses a q24 alpha-labeled path of 2K edges with its midpoint at the alpha boundary. Its selected positions carry the required zero and maximum; reversing the path handles the opposite arm. The graft shifts the high path labels by Q and shifts H's labels by K−1, separating path weights Q+1,…,Q+2K from the residual weights 1,…,Q. Complementing the whole labeling turns the maximum target into zero while preserving edge differences. Alpha/graceful amalgamation and the high-side interval shift are established operations; Barrientos (2022) describes this operation-level overlap. The reviewed comparison does not determine whether earlier work contains the exact four prescribed-position conclusion.
The author Lean 4.34.0 proof, a separate mathematical check, and a separate copied-source Lean replay passed as internal project checks; they are not external peer review. Worldwide priority remains unknown.
For every t≥1, K=23+24t, d₀=20+22t, and d₁=21+22t. For every finite indexed graph H with Q edges and conventional graceful labeling g:V(H)→{0,…,Q}, and every actual vertex r with g(r)=0, form X by attaching two separately named K-edge paths at r and retaining H. For each side s∈{left,right} and each d∈{d₀,d₁}, X has a conventional graceful labeling f with f(v(s,d))=0. The four side-and-depth choices are separate existence statements; their labelings may differ.
Formal verification
Formally verified within the stated scope
Review status
Internal check
Evidence
Author Lean proof, separate mathematical QA, and copied-source Lean replay passed; these are internal project checks.
Scope
For each t≥1, K=23+24t and depths d₀=20+22t, d₁=21+22t. For any finite indexed graph H with Q edges and a conventional graceful labeling g:V(H)→{0,…,Q}, choose an actual vertex r with g(r)=0 and attach two separately named K-edge paths at r, retaining H. For each of the two sides and each specified depth, a conventional graceful labeling of the graft assigns zero to that vertex. The four choices are separate existence claims; the labeling may vary.
Assumptions
Conventional graceful labeling means an injective vertex labeling into 0..|E| whose edge differences are exactly 1..|E|; it is not required to be onto the vertex-label interval. H is finite and indexed; a selected actual root r has label zero. The two new arms are distinct K-edge paths and every old vertex and edge of H is retained.
Toolchain
Lean 4.34.0; author proof and separate copied-source replay passed.
Reviewer
Internal project checks; no external or peer review is claimed.
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Contributions & AI assistance
Contributions
Jordi Gartner publishes this work through Beedbyte.
AI assistance
AI tools assisted with proof development and Lean/checker code, targeted comparison of earlier grafting literature, and translation drafts.
Beedbyte
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