A note on 1-2-3 and 1-2 Conjectures for 3-regular graphs

TL;DR

This paper confirms the 1-2 Conjecture for 3-regular graphs and demonstrates that all such graphs can be distinguished with a neighbor sum coloring using only 4 colors, advancing understanding of graph coloring conjectures.

Contribution

The paper proves the 1-2 Conjecture for 3-regular graphs and establishes a 4-color neighbor sum distinguishing coloring for these graphs.

Findings

Confirmed 1-2 Conjecture for 3-regular graphs.

Established a 4-color neighbor sum distinguishing coloring for 3-regular graphs.

Provided positive evidence for the 1-2-3 Conjecture in the context of 3-regular graphs.

Abstract

The 1-2-3 Conjecture, posed by Karo\'{n}ski, {\L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The 1-2 Conjecture states that if vertices also receive colors and the vertex color is added to the sum of its incident edges, then adjacent vertices can be distinguished using only $\{ 1,2\}$. In this paper we confirm 1-2 Conjecture for 3-regular graphs. Meanwhile, we show that every 3-regular graph can achieve a neighbor sum distinguishing edge coloring by using 4 colors, which answers 1-2-3 Conjecture positively.