Every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx+1}$

TL;DR

This paper proves that every tree with n edges can decompose certain complete bipartite and complete graphs using a novel algebraic approach, confirming the graceful tree conjecture and introducing new algebraic properties.

Contribution

It introduces a new algebraic method using $eta$-labelings and polynomial techniques to decompose graphs with trees, proving the graceful tree conjecture.

Findings

Every tree on n edges decomposes $K_{nx,nx}$ and $K_{2nx+1}$.

Proof of the graceful tree conjecture (1967).

Introduction of algebraic properties from decomposition results.

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.