Proving two graphs have the same chromatic number Let $G= (V,E)$ be a graph, and let $G'= (V',E')$ be a copy of $G$.  That is, for each $v ∈ V$ there  is  a  corresponding $v' ∈ V'$ and  for  each  edge  $(u,v)∈E$ there  is  a  corresponding edge $(u',v')∈E'$.  Construct a graph $G\widehat{}$ by drawing an edge from each $v∈V$ to its corresponding $v'∈V'$.  Prove that $\chi (G\widehat{}) =\chi (G)$.
My work: I sketched a few graphs such that they could be colored using two color and drew $G$ and $G'$ for each. By construction it's apparent that $G\widehat{}$ could also be colored using two colors. But i'm having a hard time extending this to more than two colors and actually writing down a proof, as a picture doesn't really count as proof.  

If I'm understanding your question correctly, the new graph $\hat{G}$ is just two copies of $G$ with the corresponding vertices connected right? If so, just think about permuting the colors on the second copy. Explicitly, say you colored $V$ with the colors $\{1, 2, \ldots k\}$. Then for $v'\in V'$, if the corresponding $v\in V$ has color $i$ color $v'$ with $i+1 \mod k$. 
