Year: 2026
Type: preprint
Abstract: We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\sim G(n,1/2)$, $χ_m(G)\le3$ with high probability, improving the previously known bound of five. Here $χ_m(G)$ denotes the multiset chromatic number of $G$, the minimum number of parts in a vertex partition whose neighbor-count vectors distinguish every pair of adjacent vertices. In fact, the three-part bound holds for every fixed $0.185
Source: arXiv (Cornell University)
Authors: Ahadi, Arash, Alipour, Sharareh
Institution: Khatam University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213999076