List Coloring Of Edges And Faces And Group Coloring

Let G be a graph,a k-list coloring of edges and faces of a graph G satisfies: for any e € E(G), f ∈ F(G), where L(e) and L(f) are any list of edges and faces and |L(e)| = |L(f)| = k, we can select a color in their lists such that any two elements incident or adjacent in E(G) ∪ F(G) are assigned distinct color. G is k-list colorable of edges and faces if there exists a k-list coloring of edges and faces of G. The choice number of edges and faces is the minimal k for which G is k-list colorable of edges and faces, denote by x_l^ef(G). In this paper, we consider the choice number of edges and faces of Halin graph and series parallel graph.A-coloring (or (A,f)—coloring ) of G under the orientation D is a function c : V(G) → A, where G = (V, E) be a graph and A a non-trivial Abelian group, denote by F(G, A) the set of all functions f : E(G) → A, such that for every arc e = (x,y) ∈ E(G), c(x) — c(y) ≠(e). The group chromatic number of a graph G is defined to be the minimum number m for which G is A — colorable for any group of order at least m under the orientation D. The group chromatic number of graph G under the orientation D is simply denoted by X₁(G).In this paper, we consider the group coloring of Halin graph. Some unsolved problems which we are interested in are listed in the last part of this paper.