A proof of the Kotzig-Ringel-Rosa Conjecture

TL;DR

This paper proves the long-standing Kotzig-Ringel-Rosa conjecture by demonstrating that every tree can be gracefully labeled, using a novel functional reformulation and a composition lemma in graph theory.

Contribution

The paper introduces a new proof of the conjecture through a functional reformulation and a composition lemma, establishing that all trees admit graceful labelings.

Findings

Proof of the Kotzig-Ringel-Rosa conjecture.

Every tree admits a graceful labeling.

New functional approach to graph labeling problems.

Abstract

In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference of labels assigned to its endpoints. The Kotzig-Ringel-Rosa conjecture asserts that every tree admits a graceful labeling. We provide a proof of this long standing conjecture via a functional reformulation of the conjecture and a composition lemma.