On odd-graceful coloring of graphs

For a graph G(V, E) which is undirected, simple, and finite, we denote by |V | and |E| the cardinality of the vertex set V and the edge set E of G, respectively. A graceful labeling f for the graph G is an injective function f : V → {0, 1, 2, . . . , |E|} such that {|f(u) − f(v)| : uv ∈ E} = {1, 2, . . . , |E|}. A graph that has a graceful-labeling is called graceful graph. A vertex (resp. edge) coloring is an assignment of color (positive integer) to every vertex (resp. edge) of G such that any two adjacent vertices (resp. edges) have different colors. A graceful coloring of G is a vertex coloring c : V → {1, 2, . . . , k}, for some positive integer k, which induces edge coloring |c(u) − c(v)|, uv ∈ E. If c also satisfies additional property that every induced edge color is odd, then the coloring c is called an odd-graceful coloring of G. If an odd-graceful coloring c exists for G, then the smallest number k which maintains c as an odd-graceful coloring, is called odd-graceful chromatic number for G. In the latter case we will denote the oddgraceful chromatic number of G as Xog(G) = k. Otherwise, if G does not admit oddgraceful coloring, we will denote its odd-graceful chromatic number as Xog(G) = ∞. In this paper, we derived some facts of odd-graceful coloring and determined odd-graceful chromatic numbers of some basic graphs.