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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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