Anti-van der Waerden Numbers of Graph Products with Trees

Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G)\to [1,\dots ,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\{v_1,\dots ,v_j\}$ such that $dist(v_i,v_{i+1})=d$ for $1\le i<j$. An arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of $G$ and is denoted $aw(G,3)$. It is known that $3\le aw(G\mathrm{\square }H,3)\le 4$. Here we determine exact values $aw(T\mathrm{\square }{T}^{\mathrm{\prime }},3)$ for some trees $T$ and $T^{\mathrm{\prime }}$, determine $aw(G\mathrm{\square }T,3)$ for some trees $T$, and determine $aw(G\mathrm{\square }H,3)$ for some graphs $G$ and $H$.