Sharp Bounds and Precise Values for the $N_i$-Chromatic Number of Graphs

TL;DR

This paper establishes sharp bounds and exact values for the $N_i$-chromatic number of graphs, relating it to parameters like vertex cover, degree, and diameter, advancing understanding of graph coloring constraints.

Contribution

It provides the first sharp bounds and exact values for the $N_i$-chromatic number based on key graph parameters.

Findings

Sharp bounds for $t_i(G)$ in terms of vertex cover, degree, and diameter.

Exact values of $t_i(G)$ determined for specific graph classes.

Enhanced understanding of $N_i$-vertex coloring constraints.

Abstract

Let $G$ be a connected undirected graph.~A vertex coloring $f$ of $G$ is an $N_i$-vertex coloring if for each vertex $x$ in $G$, the number of different colors assigned to $N_G(x)$ is at most $i$.~The $N_i$-chromatic number of $G$, denoted by $t_i(G)$, is the maximum number of colors which are used in an $N_i$-vertex coloring of $G$. In this paper, we provide sharp bounds for $t_i(G)$ of a graph $G$ in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for $t_i(G)$ in some cases.