So a coloring of \(G+e\) with the smallest possible number of colors is a best coloring of \(G\) in which \(v\) and \(w\) have different colors, that is, \(\chi(G+e)\) is the smallest number of colors...So a coloring of \(G+e\) with the smallest possible number of colors is a best coloring of \(G\) in which \(v\) and \(w\) have different colors, that is, \(\chi(G+e)\) is the smallest number of colors needed to color \(G\) so that \(v\) and \(w\) have different colors.
