Opuscula Math. 38, no. 3 (2018), 379-393
https://doi.org/10.7494/OpMath.2018.38.3.379

Opuscula Mathematica

# On expansive and anti-expansive tree maps

Sergiy Kozerenko

Abstract. With every self-map on the vertex set of a finite tree one can associate the directed graph of a special type which is called the Markov graph. Expansive and anti-expansive tree maps are two extremal classes of maps with respect to the number of loops in their Markov graphs. In this paper we prove that a tree with at least two vertices has a perfect matching if and only if it admits an expansive cyclic permutation of its vertices. Also, we show that for every tree with at least three vertices there exists an expansive map with a weakly connected (strongly connected provided the tree has a perfect matching) Markov graph as well as anti-expansive map with a strongly connected Markov graph.

Keywords: maps on trees, Markov graphs, Sharkovsky's theorem.

Mathematics Subject Classification: 37E25, 37E15, 05C20.

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• Sergiy Kozerenko
• Taras Shevchenko National University of Kyiv, Faculty of Mechanics and Mathematics, Volodymyrska Str. 64, 01033 Kyiv, Ukraine
• Communicated by Andrzej Żak.
• Revised: 2017-12-07.
• Accepted: 2017-12-19.
• Published online: 2018-03-19. 