The achromatic number of K6 □ K7 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 $\{v1,v2\} \in E(G)$ such that $f(v_{i})=c_{i}$, $i=1,2$. The achromatic number of $G$ is the maximum number $\text{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper it is proved that $\text{achr}(K_6 \square K_7)=18$. This result finalises the determination of $\text{achr}(K_6 \square K_q)$.