Coloring the Square of Products of Cycles and Paths

The square \( G^2 \) of a graph \( G \) is a graph with the same vertex set as \( G \) in which two vertices are joined by an edge if their distance in \( G \) is at most two. For a graph \( G \), \( \chi(G^2) \), which is also known as the distance two coloring number of \( G \), is studied. We study coloring the square of grids \( P_m \Box P_n \), cylinders \( P_m \Box C_n \), and tori \( C_m \Box C_n \). For each \( m \) and \( n \) we determine \( \chi((P_m \Box P_n)^2) \), \( \chi((P_m \Box C_n)^2) \), and in some cases \( \chi((C_m \Box C_n)^2) \) while giving sharp bounds to the latter. We show that \( \chi((C_m \Box C_n)^2) \) is at most \( 8 \) except when \( m = n = 3 \), in which case the value is \( 9 \). Moreover, we conjecture that for every \( m \) (\( m \geq 5 \)) and \( n \) (\( n \geq 5 \)), we have \( 5 \leq \chi((C_m \Box C_n)^2) \leq 7 \).