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The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors. (English) Zbl 1050.05057
Hedetniemi’s conjecture states that $$\chi (G\times H)= \min (\chi (G), \chi (H))$$ for any two graphs $$G,H$$, where $$G\times H = (V(G)\times V(H)$$, $$\{\{(u_1,u_2), (v_1,v_2)\}: \{u_1,v_1\}\in E(G)$$, $$\{u_2,v_2\}\in E(H)\})$$. This paper proves $$\chi (G\times H) \geq \tfrac 12 \min (\chi _f(G), \chi _f(H)),$$ where $$\chi _f(G)$$ denotes the fractional chromatic number of $$G$$, which equals $$\min \{b/a : G$$ can be covered by $$b$$ independent sets so that every vertex is covered at least $$a$$ times}. The short and beautiful proof uses the correspondence of colorings of $$G\times H$$ with homomorphisms of $$H$$ into the exponential graph $$K^G_n$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
fractional chromatic number; Hedetniemi’s conjecture
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