Nordhaus–Gaddum problem in terms of G-free coloring

Abstract

Let [Formula: see text] be a graph. A [Formula: see text]-coloring of [Formula: see text] is a mapping [Formula: see text], if each color class induces a [Formula: see text]-free subgraph. For a graph [Formula: see text] of order at least [Formula: see text], a [Formula: see text]-free [Formula: see text]-coloring of [Formula: see text], is a mapping [Formula: see text], so that the induced subgraph by each color class of [Formula: see text], contains no copy of [Formula: see text]. The [Formula: see text]-free chromatic number of [Formula: see text], is the minimum number [Formula: see text], so that it has a [Formula: see text]-free [Formula: see text]-coloring, and denoted by [Formula: see text]. Suppose that [Formula: see text] be a family of graphs, we say a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring, if there exists a map [Formula: see text], such that each color class [Formula: see text] does not contain any members of [Formula: see text]. In this paper, we give some bounds and attributes on the [Formula: see text]-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 [Formula: see text]-free chromatic number of a graph.