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Paths with two blocks in $$n$$-chromatic digraphs. (English) Zbl 1119.05049
Let $$k+l=n-1 \geq 3$$ and let $$D$$ be an $$n$$-chromatic digraph. Proving a conjecture of El-Sahili, the authors show that $$D$$ contains a $$P(k,l).$$ (Here $$P(k,l)$$ is an oriented path of order $$k+l+1$$ starting with $$k$$ forward arcs and followed by $$l$$ backward arcs for some $$k \geq 1$$ and $$l\geq 1.$$) Several connections to related results and open problems are mentioned.

MSC:
 05C20 Directed graphs (digraphs), tournaments 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles
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References:
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