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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
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References:
 [1] Subtrees of directed graphs and hypergraphs, In Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla.), I, 28 (1980), pp. 227–239. [2] A. El, Disc Math 287 pp 151– (2004) [3] On directed paths and circuits, Theory of Graphs, and (Editors), Academic press, New York, 1968, pp. 115–118. [4] F., Graphs Combinat 19 pp 101– (2003) [5] F., J Graph Theory 35 pp 244– (2000) [6] F., J Combinat Theory Ser B 78 pp 243– (2000) [7] B., Rev Française Automat Informat Recherche Opérationelle Sér Rouge 1 pp 127– (1967) [8] M., Colloquia Mathematica Societatis János Bolyai 37 pp 663– (1981) [9] A., Trans Amer Math Soc 296 pp 167– (1986)
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