On the Improper Dynamic Coloring of Graphs

Let G be a graph. A proper vertex k-coloring of G is called dynamic, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. If the coloring is not necessarily proper, then we call the coloring, the improper dynamic coloring. The smallest integer k such that G has an improper dynamic k-coloring is called the improper dynamic chromatic number of G and is denoted by χ id (G). Here, we study the improper dynamic chromatic number of some family of graphs. For instance, it is shown that the improper dynamic chromatic number of graphs with maximum degree at most 3, regular graphs and also claw-free graphs are at most 3. Also it is proved that the improper dynamic chromatic number of graphs containing a 1-factor is at most 3 and for the graphs containing a path-factor is at most 5. Moreover, it is shown that if G has a {K 1,1 , . . . , K 1,n }-factor, then χ id (G) ≤ 2n + 1.