Let $ G,H $ are simple finite graphs and $A = G \times H$. Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.

Let $G$ has smaller chromatic number. Experiments suggest that given a coloring $f$ of $G$ one can color $A = G \times H$. Color the vertices $(a,b)$ of $A$ with $g(a,b)=f(a), \; a \in V(G)$. Experimentally the coloring is valid.

This was verified for 1000 random graphs and for $\{ \text{Petersen graph}, K_2,K_6, C_5,\text{Star graph 6} ,\text{Random graph of order 14}, \} \times \\\\ \{\text{All graphs up to 7 vertices}\}$

This is related to Hedetniemi's Conjecture which states $ \chi(G \times H) = \min \{ \chi(G), \chi(H) \} $.

- Any counterexamples to this coloring?
- Is it possible to prove this is valid coloring for certain $G$ or $H$?
- What types of graphs are potential counterexamples?