Nordhaus–Gaddum problem in terms of G-free coloring.

Let H = (V (H) , E (H)) be a graph. A k -coloring of H is a mapping π : V (H) → { 1 , 2 , ... , k } , if each color class induces a K 2 -free subgraph. For a graph G of order at least 2 , a G -free k -coloring of H , is a mapping π : V (H) → { 1 , 2 , ... , k } , so that the induced subgraph by each color class of π , contains no copy of G. The G -free chromatic number of H , is the minimum number k , so that it has a G -free k -coloring, and denoted by χ G (H). Suppose that be a family of graphs, we say a graph H has a -free k -coloring, if there exists a map π : V (H) → { 1 , 2 , ... , k } , such that each color class V i = π − 1 (i) does not contain any members of G. In this paper, we give some bounds and attributes on the G -free chromatic number of graphs in terms of the number of vertices, maximum degree, minimum degree, and chromatic number. Our main results are the Nordhaus–Gaddum-type theorem for the -free chromatic number of a graph. [ABSTRACT FROM AUTHOR]