A proof of the Multiplicative 1-2-3 Conjecture

TL;DR

This paper proves the multiplicative version of the 1-2-3 Conjecture, demonstrating that for any connected graph with at least three vertices, edge labels 1, 2, and 3 can be assigned to distinguish adjacent vertices by their product of incident edge labels.

Contribution

It provides a proof confirming the multiplicative 1-2-3 Conjecture for all connected graphs with at least three vertices.

Findings

The conjecture holds for all connected graphs with at least 3 vertices.

Edge labels 1, 2, 3 suffice to distinguish adjacent vertices by product.

The proof resolves a longstanding open problem in graph labeling theory.

Abstract

We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kazi{\'o}w in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1,2,3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.