Attainability of the Chromatic Number of Functigraphs

Let G´ be a copy of a graph G and let f : V (G) → V(G´) be a function. A functigraph, denoted by C(G, f), is a graph with V (C(G, f )) = V(G) U V(G´) and E(C(G, f )) = E(G) U E(G´) U {uv : u ϵ V (G), v ϵ V (G´), v= f (u)}. Let χ(G) be the chromatic number of a graph. Recently, Chen et al. proved that χ(G) ≤ χ(C(G, f)) ≤ [3/2χ(G)]. In this paper we investigate the attainability of the chromatic numbers of functigraphs. We prove that for any two positive integers a and b with a ≤ b ≤ [3/2 a], there exists a graph G and function f on V (G) such that χ(G) = a and χ(C(G, f)) = b. We also characterize functions to determine the values of χ(C(G, f)) for all bipartite graphs and odd wheels. We finally provide some sufficient conditions on f to have χ(C(G, f)) = χ(G).