Attainability of the Chromatic Number of Functigraphs

Let $G'$ be a copy of a graph $G$ and let $f: V(G) \to V(G')$ be a function. A functigraph, denoted by $C(G,f)$, is a graph with $V(C(G,f)) = V(G)\cup V(G')$ and $E(C(G,f)) = E(G)\cup E(G')\cup \{uv: u\in V(G), v\in V(G'), v=f(u)\}$. Let $\chi(G)$ be the chromatic number of a graph. Recently, \textit{Chen et al.} proved that $\chi(G) \leq \chi(C(G,f)) \leq \left\lceil \frac{3}{2} \chi(G) \right\rceil.$ 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 $a \leq b \leq \left\lceil \frac{3}{2}a \right\rceil$, there exists a graph $G$ and function $f$ on $V(G)$ such that $\chi(G)=a$ and $\chi(C(G,f))=b$. We also characterize functions to determine the values of $\chi(C(G,f))$ for all bipartite graphs and odd wheels. We finally provide some sufficient conditions on $f$ to have $\chi(C(G, f)) = \chi(G)$.