1,2,3-Conjecture and 1,2-Conjecture for Sparse Graphs

TL;DR

This paper uses the discharging method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for sparse graphs with maximum average degree less than 8/3, providing new structural insights.

Contribution

First application of discharging method to these conjectures, establishing results for graphs with low maximum average degree.

Findings

Proved the 1,2,3-Conjecture for graphs with max average degree < 8/3

Proved the 1,2-Conjecture for graphs with max average degree < 8/3

Developed structure theorems and reducibility results of independent interest

Abstract

We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures have been proved, but this is the first application of discharging to them, and the structure theorems and reducibility results are of independent interest.