Anti-van der Waerden Numbers of Graph Products of Cycles

Abstract

A $k$-term arithmetic progression ($k$-AP) in a graph $G$ is a list of vertices such that each consecutive pair of vertices is the same distance apart. If $c$ is a coloring function of the vertices of $G$ and a $k$-AP in $G$ has each vertex colored distinctly, then that $k$-AP is a rainbow $k$-AP. The anti-van der Waerden number of a graph $G$ with respect to $k$ is the least positive integer $r$ such that every surjective coloring with domain $V(G)$ and codomain $\{1,2,\dots ,r\}=[r]$ is guaranteed to have a rainbow $k$-AP. This paper focuses on $3$-APs and graph products with cycles. Specifically, the anti-van der Waerden number with respect to $3$ is determined precisely for $P_m\square C_n$, $C_m\square C_n$ and $G\square C_{2n+1}$.