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A Solution to the 1-2-3 Conjecture

Ralph Keusch

Abstract

We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Luczak, and Thomason from 2004.

A Solution to the 1-2-3 Conjecture

Abstract

We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Luczak, and Thomason from 2004.
Paper Structure (4 sections, 9 theorems, 17 equations)

This paper contains 4 sections, 9 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Main ideas and proof preparations
  3. Proof of Theorem \ref{['thm:main']}
  4. Concluding remarks

Key Result

Theorem 1

Let $G=(V,E)$ be a graph without connected components isomorphic to $K_2$. Then there exists an edge-weighting $\omega:E \rightarrow \{1,2,3\}$ such that for each edge $\{v,w\} \in E$,

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4: Lemma 2 in keusch2023vertex
  • proof : Proof of Lemma \ref{['lemma:weighting']}
  • Lemma 5
  • proof
  • Definition 6
  • Lemma 7
  • ...and 8 more