Recently, I am interested in the graph polynomial of the product of cycles.
Let  $G = (V , E)$ be an undirected multi-graph with vertex set $\{1,\cdots,n\}$. The graph polynomial of $G$
is defined by
$$f_G(x_1,x_2,\cdots,x_n)=\prod_{1\leq i<j\leq n, (i,j)\in E}(x_i-x_j).$$

**Conjecture:** Let   $G $ be the Cartesian product graph $C_{2n+1}\Box  C_{2m}$, then the coefficient of $x_1^2x_2^2\cdots x_{(2n+1)(2m)}^2$ in the the graph polynomial $f_G(x_1,x_2,\cdots, x_{(2n+1)(2m)})$ is nonzero.

For $C_3\Box  C_{2n}$, the conjecture is true. See [the coefficient of a special term in the expansion of the graph polynomial][1]

This conjecture generalized the result about $C_3\Box  C_{2n}$. I think it may be true. But I have no idea about the proof on the general cases.

I hope someone could give suggestions about  the conjecture. I will appreciate it even if given some special cases for the conjecture such as $n=2,3$,etc.

  [1]: https://mathoverflow.net/questions/301898/how-to-get-the-coefficient-of-a-special-term-in-the-expansion-of-the-graph-polyn