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Abstract:We prove that every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx + 1}$ for all positive integers $x$. The said decompositions are obtained by proving that every tree admits a $\vec{\beta}$-labeling (oriented beta-labeling). Our proof employs the polynomial method by identifying trees as functions in the transformation monoid $\mathbb{Z}_n^{\mathbb{Z}_n}$. A proof of the graceful tree conjecture (1967) follows as an immediate consequence of the current result. Finally, we introduce additional algebraic properties derived from the decomposition results.
From: Parikshit Chalise [view email]
[v1]
Tue, 3 Sep 2024 15:23:47 UTC (266 KB)
[v2]
Thu, 5 Dec 2024 18:57:24 UTC (29 KB)
[v3]
Tue, 1 Sep 2026 21:47:15 UTC (1 KB) (withdrawn)
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