Abstract
The fiber product of graphs over homomorphims, a notion introduced by P. Hell (1972) is used here as a new approach to S. Hedetniemi's conjecture (1966) since it's a subgraph of the cross product. We consider here only the fiber product over colorings which are special cases of homomorphisms of graphs. We show that Khelladi's conjecture (1991): “The fiber product over colorings of two n-chromatic graphs is also an n-chromatic graph” implying trivially Hedetniemi's is true for n ≥ 3. Moreover, we propose an equivalent statement of Khelladi's conjecture using the class of graph colourings of a graph defined by El-Zahar and Sauer.
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