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Hedetniemi's conjecture

Résumé
In graph theory, Hedetniemi's conjecture, formulated by Stephen T. Hedetniemi in 1966, concerns the connection between graph coloring and the tensor product of graphs. This conjecture states that : \chi (G \times H ) = \min{\chi (G) , \chi (H)}. Here \chi(G) denotes the chromatic number of an undirected finite graph G. The inequality χ(G × H) ≤ min {χ(G), χ(H)} is easy: if G is k-colored, one can k-color G × H by using the same coloring for each copy of G in the product; symmetrically if H is k-colored. Thus, Hedetniemi's conjecture amounts to the assertion that tensor products cannot be colored with an unexpectedly small number of colors. A counterexample to the conjecture was discovered by (see ), thus disproving the conjecture in general. Known cases Any graph with a nonempty set of edges requires at least two colors; if G and H are not 1-colorable, that is, they both contain an edge, then their product also contains an edge
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