The 1-2-3 Conjecture and related problems: a survey

TL;DR

This survey reviews the current research status of the 1-2-3 Conjecture in graph theory, discussing its variants, progress, and open problems since its proposal in 2004.

Contribution

It provides a comprehensive overview of the conjecture, summarizing key results, variants, and ongoing challenges in the field.

Findings

Summary of proven cases and partial results

Overview of variants and related problems

Identification of open questions and future directions

Abstract

The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows: "If G is a graph with no connected component having exactly 2 vertices, then the edges of G may be assigned weights from the set {1,2,3} so that, for any adjacent vertices u and v, the sum of weights of edges incident to u differs from the sum of weights of edges incident to v." This survey paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.