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A proof of the Kotzig-Ringel-Rosa Conjecture

Edinah K. Gnang

TL;DR

The paper resolves the Kotzig–Ringel–Rosa conjecture by reframing graceful labeling as a functional problem on the transformation monoid $\mathbb{Z}_{n}^{\mathbb{Z}_{n}}$ and establishing a Composition Lemma that transfers graceful labelings to suitable iterates. It introduces the Graceful Expansion Theorem and a determinantal certificate within a Lagrange-Groebner algebraic framework to certify graceful labelings. The main result shows that every tree on $n$ vertices has a graceful labeling, yielding a non-asymptotic decomposition of $K_{2n-1}$ into edge-disjoint copies of any $n$-vertex tree. By blending combinatorics, algebra, and group actions, the work provides a novel route to proving KRR with potential implications for related graph-decomposition 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.

A proof of the Kotzig-Ringel-Rosa Conjecture

TL;DR

The paper resolves the Kotzig–Ringel–Rosa conjecture by reframing graceful labeling as a functional problem on the transformation monoid and establishing a Composition Lemma that transfers graceful labelings to suitable iterates. It introduces the Graceful Expansion Theorem and a determinantal certificate within a Lagrange-Groebner algebraic framework to certify graceful labelings. The main result shows that every tree on vertices has a graceful labeling, yielding a non-asymptotic decomposition of into edge-disjoint copies of any -vertex tree. By blending combinatorics, algebra, and group actions, the work provides a novel route to proving KRR with potential implications for related graph-decomposition 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.
Paper Structure (7 sections, 16 theorems, 212 equations, 2 figures)

This paper contains 7 sections, 16 theorems, 212 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional formulation
  4. Preliminaries
  5. Determinantal Certificate
  6. The transformation monoid $\mathbb{Z}_{n}^{\mathbb{Z}_{n}}$ and the composition lemma.
  7. The Graceful Labeling Theorem

Key Result

Theorem 1

Every tree admits a graceful labeling.

Figures (2)

  • Figure 2.1: $f\left(0\right)=0,\,f\left(1\right)=3,\,f\left(2\right)=3,\,f\left(3\right)=0,\,f\left(4\right)=0,\,f\left(5\right)=0$
  • Figure 3.1: $f\left(0\right)=0,\,f\left(1\right)=0,\,f\left(2\right)=1,\,f\left(3\right)=2$

Theorems & Definitions (43)

  • Theorem 1
  • Definition 2
  • Remark
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 6: Graceful Expansion Theorem
  • proof
  • Example 7
  • Definition 8
  • ...and 33 more