The categorical product GxH has vertex set the cartesian product of V(G) and V(H), and edge set all the pairs (x,y)(x',y') for which xx' is an edge of G and yy' is an edge of H.
It is an easy observation, using projections on G or on H, that
the chromatic number of GxH is at most the minimum of the chromatic
numbers of G and H.
We let g(k) to be the minimum chromatic number one can
obtain by the categorical product of two k-chromatic graphs.
The conjecture asserts that g(k)=k.
Very few is known about g(k), it is not even known if it goes to infinity. But the following was proved:
Either g is bounded by nine or goes to infinity.