Separate zero placements at every interior depth of two rooted 119-edge arms

Let H be any finite graph with Q edges, a specified root o, and a supplied conventional graceful labeling g. Its vertex labels are distinct values in 0,…,Q, its edge absolute differences are exactly 1,…,Q, and g(o)=0. Retain every vertex and edge of H and attach two separately named 119-edge arms at o, identifying o with index 119 of a 238-edge path. For each depth d=1,…,118 on either new arm, the entire graft has a conventional graceful labeling assigning zero to that selected vertex. Each target has its own labeling; the vertex labels lie in 0,…,Q+238 and the edge differences are exactly 1,…,Q+238.

The supplied root-zero labeling is a premise. H need not be connected, acyclic, a tree, alpha-labeled or onto-labeled; Q=0 is allowed. The theorem does not prescribe zero at the shared root, either new tip at depth 119, or old vertices of H, and does not give one labeling with several zero targets.

The source also proves a separate near-tip family. For every integer r≥1, set K=30·4^r−1. With the same supplied root-zero H premise, each depth K−8,…,K−1 on either newly attached K-edge arm has its own conventional graceful labeling of the entire graft, with edge span 2K+Q. These eight depths arise from four source theorem pairs. This does not establish all interior depths for r>1 or every odd arm length.

The fixed K119 proof combines the retained 1–114 theorem with terminal-seed proofs for 115–116 and 117–118. Certified alpha paths are attached using established alpha/graceful vertex amalgamation, with label complementation when the selected path vertex carries the maximum. Earlier sources for this graft include Huang–Kotzig–Rosa, as stated in Panpa, Imnang and Wasuanankul (2025, Theorem 2.5) and Shan–Zhong (2026, Lemma 1). The endpoint-matched path concatenation is also an older operation, appearing in Adamaszek (Lemma 1) and the proof of Hicks–Ollis–Schmitt (Lemma 4.5).

Two applications of Shan–Zhong (Theorem 2) already give the arbitrary-H depth-1 and depth-2 cases on either arm; Luiz–Campos–Richter (2017, Theorem 14) covers a path with one central leaf. These operations and subcases are credited as prior work. Whether older results imply the complete K119 range 1–118 remains unresolved, including the full Cattell (2007) alpha/path construction. Whether the complete depth-1–118 statement was already known is unknown.