On r− dynamic coloring of the gear graph families

T Deepa,Dafik Dafik,M Venkatachalam

doi:10.22199/issn.0717-6279-2021-01-0001

Abstract

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, d(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r−dynamic chromatic number of the middle, central and line graphs of the gear graph.

Highlights

Graphs in this paper are simple and finite
An upper bound for the dynamic chromatic number of a d-regular graph G in terms of χ(G) and the independence number of G, α(G), was introduced in [7]
Taherkhani gave in [15] an upper bound for χ2(G) in terms of the chromatic number,lthe maximu3m 3degree ∆ ́ ́amnd the minimum degree δ. i.e., χ2(G) − χ(G) ≤ (∆e)/δlog 2e ∆2 + 1

Summary

Introduction

Graphs in this paper are simple and finite. For undefined terminologies and notations see [5, 17]. The r-dynamic chromatic number of a graph G, written χr(G), is the minimum k such that G has an r-dynamic proper k-coloring. An upper bound for the dynamic chromatic number of a d-regular graph G in terms of χ(G) and the independence number of G, α(G), was introduced in [7]. Taherkhani gave in [15] an upper bound for χ2(G) in terms of the chromatic number,lthe maximu3m 3degree ∆ ́ ́amnd the minimum degree δ. N.Mohanapriya et al [11, 12] studied the dynamic chromatic number for various graph families It was proven in [13] that the r-dynamic chromatic number of line graph of a helm graph Hn. In this paper, we study χr(G), when 1 ≤ r ≤ ∆. We find the r- dynamic chromatic number of the middle, central and line graphs of the gear graph

Preliminaries

Main Theorem

Result