The achromatic number of $K_6\square K_q$ equals $2q+3$ if $q\ge41$ is odd

Abstract

Let $G$ be a graph and $C$ a finite set of colours. A vertex colouring $f:V(G)\to C$ is complete provided that for any two distinct colours $c_1,c_2\in C$ there is $v_1v_2\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper it is proved that if $q\ge41$ is an odd integer, then the achromatic number of the Cartesian product of $K_6$ and $K_q$ is $2q+3$.