On expansive and anti-expansive tree maps

Abstract

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.

Vertices with the second neighborhood property in Eulerian digraphs

“…Ever since, further results have primarily consisted of applications of local median orders to tournaments missing a well-defined structure such as a generalized star or a subset of the arc set of a smaller tournament [10,6], or have dealt strictly with dense digraphs [8]. Markov graphs are known to satisfy the Second Neighborhood Conjecture [13]. Progress has also been made on quasi-transitive graphs [14].…”