The achromatic number of $K_6\square K_7$ is $18$

Abstract

A vertex colouring $f:V(G)\to C$ 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 $f(v_i)=c_i$, $i=1,2$. 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 $\mathrm{achr}(K_6\square K_7)=18$. This result finalises the determination of $\mathrm{achr}(K_6\square K_q)$.