SOME RESULTS IN LIST G-FREE COLOURINGS OF GRAPHS

For a given graph H and a graphical property P, the conditional chromatic number χ(H, P) of H, is the smallest num_x0002_ber k, sush that V (H) can be decomposed into sets V1, V2, . . . , Vk, in which H[Vi] satisfies the property P, for each 1 ≤ i ≤ k. When property P is that each color class contains no copy of G, we write χG(H) instead of χ(G, P), which is called the G-free chromatic number. Due to this, we say H has a k-G-free coloring if there is a map c : V (H) −→ {1, . . . , k}, such that each of the color classes of c is G-free. Assume that for each vertex v of a graph H is assigned a set L(V ) of colors, called a color list. Set g(L) = {g(v) : v ∈ V (H)}, that is the set of colors chosen for the vertices of H under g. An L-coloring g is called G-free, such that, I:g(v) ∈ L(v), for any v ∈ V (H), II: H[Vi] is G-free for each i = 1, 2, . . . , L. If there exists an L-coloring of H, then H is called L-G-free-colorable. A graph H is said to be k-G-free-choosable if there exists an L-coloring for any list-assignment L satisfying |L(V )| ≥ k for each v ∈ V (H), and H[Vi] be G-free for each i = 1, 2, . . . , L. In this article, we determine some upper bounds for χLG(H), in term of the ∆(H), |V (H)| and δ(G). Also, we show that χLG(H) = χG(H) for some graph H and G.