On the achromatic number of the Cartesian product of two complete graphs

TL;DR

This paper determines the achromatic number of the Cartesian product of a complete graph and a complete bipartite graph for infinitely many cases, using properties of finite projective planes.

Contribution

It provides new exact values for the achromatic number of specific graph products involving complete graphs and finite projective planes.

Findings

Achromatic number of $K_{r^2+r+1} \, \square \, K_q$ determined for infinitely many $q$.

Utilizes properties of finite projective planes to derive results.

Advances understanding of graph coloring in Cartesian products.

Abstract

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any $c_1,c_2\in C$ with $c_1\ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. Let $G_1\square G_2$ denote the Cartesian product of graphs $G_1$ and $G_2$. In the paper $\mathrm{achr}(K_{r^2+r+1}\square K_q)$ is determined for an infinite number of $q$s provided that $r$ is a finite projective plane order.