A note on 1-2-3 and 1-2 Conjectures for 3-regular graphs
Combinatorics
2021-07-02 v1
Abstract
The 1-2-3 Conjecture, posed by Karo\'{n}ski, {\L}uczak and Thomason, asked whether every connected graph different from can be 3-edge-weighted so that every two adjacent vertices of 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 . 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.
Cite
@article{arxiv.2107.00424,
title = {A note on 1-2-3 and 1-2 Conjectures for 3-regular graphs},
author = {Jing-zhi Chang and Chao Yang and Zhi-xiang Yin and Bing Yao},
journal= {arXiv preprint arXiv:2107.00424},
year = {2021}
}