On the Chromatic Number of Cayley Tables of Dihedral Groups

Abstract

A coloring of a Latin square $L$ is an assignment of $k$ colors to its cells such that no two cells sharing a row, a column, or a symbol receive the same color. The smallest $k$ for which such a coloring exists is called the \emph{chromatic number} of $L$, denoted by $\chi (L)$. The Cayley table of a finite group $G$ of order $n$ is a Latin square, which we denote by $L_G$. It is known that $\chi (L_G)\ne n+1$, so either $\chi (L_G)=n$ or $\chi (L_G)\ge n+2$. Moreover, $\chi (L_G)=n$ if and only if every Sylow $2$-subgroup of $G$ is trivial or non-cyclic, which completely characterizes the finite groups whose Cayley tables attain the lower bound $n$. In this paper, we consider the dihedral group $D_n$ of order $2n$, where $n$ is odd. Since $D_n$ has a cyclic Sylow $2$-subgroup, it follows that $\chi (L_{D_n})\ge 2n+2$. We prove that this bound is sharp, that is, $\chi (L_{D_n})=2n+2$.