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About paths with two blocks. (English) Zbl 1121.05065
Summary: The function $$f(n)$$ is defined to be the smallest integer such that any $$f(n)$$-chromatic digraph contains all paths with two blocks $$P(k,j)$$ with $$k + j = n - 1$$. A. El Sahili [Discrete Math. 287, No. 1–3, 151–153 (2004; Zbl 1050.05072)] conjectured that $$f(n) = n$$. We prove in this paper that $$f(n) \leq n + 1$$. Our argument yields a very short and direct proof of the Gallai-Roy result about directed paths. We also treat the problem under some supplementary conditions. One of them is used to give a simple proof of a result of M. Saks and V. T. Sós [Colloq. Math. Soc. János Bolyai 37, No. 2, 663–674 (1984; Zbl 0566.05032)] about claws.

##### MSC:
 05C38 Paths and cycles 05C20 Directed graphs (digraphs), tournaments 05C15 Coloring of graphs and hypergraphs
##### Keywords:
digraphs; chromatic number; paths and cycles
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##### References:
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