The achromatic number of the Cartesian product of $K_6$ and $K_q$

TL;DR

This paper determines the achromatic number of the Cartesian product of the complete graph $K_6$ with $K_q$ for specific ranges of $q$, expanding understanding of graph colorings in product graphs.

Contribution

It provides exact values of the achromatic number for $K_6 imes K_q$ for certain ranges of $q$, filling gaps in the existing literature.

Findings

Achromatic number computed for $8 \\le q \\le 40$ and even $q \\ge 42$.

Established new bounds and exact values for the achromatic number in these cases.

Enhanced understanding of colorings in Cartesian product graphs.

Abstract

Let $G$ be a graph and $C$ a finite set of colours. A vertex colouring $f:V(G)\to C$ is complete if for any pair of distinct colours $c_1,c_2\in C$ one can find 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 defined to be the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper $\mathrm{achr}(K_6\square K_q)$ is determined for any integer $q$ such that either $8\le q\le40$ or $q\ge42$ is even.