On r-dynamic chromatic number of graphs

Abstract

An r-dynamic k-coloring of a graph G is a proper vertex k-coloring such that the neighbors of any vertex v receive at least min{r, deg(v)} different colors. The r-dynamic chromatic number of G, χ r (G), is defined as the smallest k such that G admits an r-dynamic k-coloring. In this paper we introduce an upper bound for χ r (G) in terms of r, chromatic number, maximum degree and minimum degree. In 2001, Montgomery [9] conjectured that, for a d-regular graph G, χ 2 (G) − χ(G) ≤ 2. In this regard, for a d-regular graph G, we present two upper bounds for χ 2 (G) − χ(G), one of them, ⌈5.437 log d + 2.721⌉, is an improvement of the bound 14.06 log d + 1, proved by Alishahi (2011) [2]. Also, we give an upper bound for χ 2 (G) in terms of chromatic number, maximum degree and minimum degree.

Dynamic Chromatic Number of Bipartite Graphs

“…Afterwards, it was shown in [8] that for every k-regular graph G with no induced [8] gave a negative answer to this conjecture. Recently, Taherkhani in [21]…”

On r− dynamic coloring of the gear graph families

“…In fact, it was proved that χ d (G) ≤ χ(G) + 2log 2 α(G) + 3. Taherkhani gave in [15] an upper bound for χ 2 (G) in terms of the chromatic number, the maximum degree ∆ and the minimum degree δ. i.e., χ…”

On Dynamic Coloring of Web Graph

“…The dynamic chromatic number = ( ) is the minimum k for which G has a dynamic k-coloring. The dynamic chromatic number, χd(G), has been investigated in several papers, see, (1,2,3,4,5,8,13,14). In 2001 Montgomery (13) conjectured that for a regular graph G, (G) − ( ) ≤ 2.…”