# A conjecture on the coefficient of a special term in the expansion of the graph polynomial?

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

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.