The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

$$ \chi _{f} ( G \times H ) \geqslant \frac{1} {4} \cdot \min \{ \chi _{f} ( G ),\chi _{f} ( H ) \} $$

holds for all directed graphs G and H. Similar bounds for the usual chromatic number seem to be much harder to obtain: It is still not known whether there exists a number n such that χ(G×H) ≥ 4 for all directed graphs G, H with χ(G) ≥ χ(H) ≥ n. In fact, we prove that for every integer n ≥ 4, there exist directed graphs G n , H n such that χ(G n ) = n, χ(H n ) = 4 and χ(G n ×H n) = 3.

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Correspondence to Claude Tardif.

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Tardif, C. The Fractional Chromatic Number Of The Categorical Product Of Graphs. Combinatorica 25, 625–632 (2005). https://doi.org/10.1007/s00493-005-0038-y

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Mathematics Subject Classification (2000):

  • 05C15