The achromatic number of $K6 \square K7$ is 18

A vertex colouring $ƒ: V(G) \rightarrow C$ of a graph $G$ is complete if for any two distinct colours $c_{1},c_{2} \in C$ there is an edge $\{v_{1} ,v_{2}\} \in E(G)$ such that $ƒ(v_{i}) = c_{i}, i = 1, 2$. The achromatic number of G is the maximum number achr(G) of colours in a proper complete vertex colouring of G. In the paper it is proved that achr$(K_{6} \square K_{7})$ = 18. This result finalises the determination of achr$(K_{6} \square K_{q})$.