BeedbyteBeedbyte / Publications
Publications
Research notes, manuscripts, software and data. Each publication lists its versions and any change notes.
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A certificate-and-composition proof of 0-rotatability for S(k^n,1^m), k in {3,5,7,9,11}, n >= 2, m >= 0. Established ingredients and prior subfamilies are credited; publication priority remains unresolved.
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For every even K≥20, attach two K-edge arms to the chosen zero-labeled root of any finite graph H with a supplied conventional graceful labeling. On either new arm, each depth 2≤d≤D8(K) separately admits a conventional graceful labeling with zero at that vertex. The guaranteed depth interval grows without bound.
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For every integer k≥19, all n≥2,m≥0 and any selected long arm of S(k^n,1^m), each depth 2≤d≤D8(k) can receive zero in a graceful labeling. Separate requests may use separate labelings. The explicit sufficient prefix includes both length parities and has limiting proportion 10/11; no optimality or historical priority is claimed.
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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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For k=23+24t, n≥2 and m≥0, every selected named long-arm vertex at depth 17+20t through 20+22t can be zero in its own graceful labeling for every integer t≥0. For t≥1, the same holds at the additional depth 21+22t. For t≥2, it also holds at depth 22+22t. Consequently, for t≥5 the q24/D8 prefix and q48 point combine to give all depths 1≤d≤22+22t.
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Given a finite graph with a supplied graceful labeling that assigns zero to a chosen root, attach two new 119-edge arms there. Every interior depth 1–118 on either arm can receive zero in its own labeling. The source also proves eight specified near-tip depths for K=30·4^r−1, r≥1.
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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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A single 1 at position 0 and a right-infinite 1010… pattern starting at 0 give the same center sequence, with all other cells initially 0. Their evolution also agrees everywhere to the left.
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For every integer m≥1 and every prescribed vertex of S(7,7,7,1^m), there exists a graceful labeling that assigns zero to that vertex. Five base certificates, four complements, and a leaf-insertion lemma prove the result; publication priority remains unresolved.
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For every n ≥ 2 and m ≥ 0, each actual vertex of S(23^n,1^m) can be assigned zero in a graceful labeling, with a separate labeling for each selected vertex.
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For every n ≥ 2 and m ≥ 0, every actual vertex of S(25^n,1^m) can receive zero in a graceful labeling, with a separate labeling for each selected vertex.
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For every n ≥ 2 and m ≥ 0, each actual vertex of S(47^n,1^m) can be the unique zero in a graceful labeling; a separate labeling may be used for each target.
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For every n ≥ 2 and m ≥ 0, each actual vertex of S(71^n,1^m) can be the unique zero in a graceful labeling; a separate labeling may be used for each target.
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For every n≥2 and m≥0, each vertex of S(95^n,1^m)—the hub, any of its m original leaves, or any position on any named 95-edge arm—can be assigned zero in its own graceful labeling.
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For every even k ≥ 2, n ≥ 2, and m ≥ 0, each named vertex of S(k^n,1^m) can receive zero in a graceful labeling, with a separate labeling for each chosen vertex.
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For every integer m >= 1, both S(3,3,3,1^m) and S(4,4,4,1^m) are 0-rotatable. Eleven explicit base labelings and one threshold-insertion lemma give a written proof for all m.