Let $G$ be an undirected $6 \times 6$ lattice graph. The $36$ vertices of $G$ are each randomly colored with one of $5$ colors with equal probability. Such a coloring is called "successful" if and only if there exists at least one nontrivial monochromatic cycle. What is the probability that such a random coloring on $G$ is successful?
For a simpler version of this question, suppose we are interested in the probability that $G$ contains a monochromatic $4$-cycle--i.e., a $2 \times 2$ square. This should be more tractable, I think. It is very easy to compute explicit probabilities in closed form for $2 \times 2$ and $3 \times 3$ lattices. Alternatively, we could consider fewer colors, say just $2$.
As a side question, has this sort of problem been investigated in the mathematical literature? Generalizations that provide bounds on the number of such graphs?
(Bonus points if you know where the inspiration for this question comes from.)