A complete coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two distinct colors \( i \) and \( j \) used in the coloring, there exist adjacent vertices of \( G \) colored \( i \) and \( j \). The maximum positive integer \( k \) for which \( G \) has a complete \( k \)-coloring is the achromatic number \( \psi(G) \) of \( G \).
A Grundy coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two colors (positive integers) \( i \) and \( j \) with \( i < j \), every vertex colored \( j \) has a neighbor colored \( i \). The maximum positive integer \( k \) for which a graph \( G \) has a Grundy \( k \)-coloring is the Grundy number \( \Gamma(G) \). Thus, \( 2 \leq \chi(G) \leq \Gamma(G) \leq \psi(G) \) for every nonempty graph \( G \). It is shown that if \( a, b, \) and \( c \) are integers with \( 2 \leq a \leq b \leq c \), then there exists a connected graph \( G \) with \( \chi(G) = a \), \( \Gamma(G) = b \), and \( \psi(G) = c \) if and only if \( a = b = c = 2 \) or \( b \geq 3 \).