On the inducibility of small trees
CITATION: Dossou-Olory, Audace A. V. & Wagner, S. 2019. On the inducibility of small trees. Discrete Mathematics and Theoretical Computer Science, 21:4, #14, doi:0.23638/DMTCS-21-4-13.
The original publication is available at https://dmtcs.episciences.org
The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Sz´ekely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be ‘close’ to 1/4. We can make this precise by showing that 0.24707 . . . ≤ J(A5) ≤ 0.24745 . . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.