Achromatic number of \(K_5\times K_n\) for large \(n\) (Q5936062)

The achromatic number of a graph \(G\) is the maximum number of colors in a proper vertex coloring of \(G\) such that vertices of any two distinct color classes are joined by at least one edge. In this paper it is shown that the achromatic number of the Cartesian product of \(K_{5}\) and \(K_{n}\) is equal to \(2n-1\) if \(n=25\); \(2n-2\) if \(26\leq n\leq 28\);\( \lfloor 3n/2\rfloor +12\) if \(29\leq n\leq 36\); \(\lfloor 5n/3\rfloor +6\) if \(37\leq n\leq 42\); \(\lfloor 9n/5 \rfloor \) if \(n\geq 43 \).