Nahn Nguyen

Ak-edge-weighting of a graphGis a mapw:E(G)→{1,···, k}.Everyk-edge-weightingwinduces a vertex coloring onGbyadding the weight of all the edges at any vertex. In 2004 Karon-ski, Luczak and Thomason conjectured that for any connectedgraphGwith at least 3 vertices, there exists a 3-edge-weightingthat induces a proper vertex coloring. This conjecture cameto be known as the 123 Conjecture. To this day it remains aconjecture; though progress has been made in the form of the12345 Theorem and other partial results.This talk will focus on variants of the 123 Conjecture includingbut not limited “total colors” and “twin edge colorings”. Ourresults on the twin edge coloring problem will be presented alongwith open problems and directions for future research.This talk is accessible to any undergraduate student with anunderstanding of modular arithmetic.