  For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory,
the following famous conjecture posed by Hedetniemi has been studied: For two
graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G
\times H$ is the tensor product of $G$ and $H$. In this paper, we give a
reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals
of polynomial rings, and we demonstrate computational experiments for partial
solutions of Hedetniemi's conjecture along such a strategy using Gr\"{o}bner
basis.
