Recently, I am interested in the polynomial 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

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.

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