On Dynamic Coloring of Graphs

TL;DR

This paper investigates upper bounds on the dynamic chromatic number of graphs, especially regular graphs, and introduces bounds for the dynamic list chromatic number, advancing understanding of graph coloring constraints.

Contribution

It provides new upper bounds for the dynamic chromatic and list chromatic numbers of regular graphs, including a logarithmic bound related to degree.

Findings

Bound: (G)  \u2212  (G)  + c \u2212 1 for some constant c

Upper bounds for the dynamic list chromatic number of regular graphs

Establishment of a relationship between degree and dynamic coloring parameters

Abstract

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such that for every $k$-regular graph $G$, $\chi_d(G)\leq \chi(G)+ c\ln k +1$. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.