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Graphs and Combinatorics

, Volume 35, Issue 1, pp 303–320 | Cite as

The Minimum Asymptotic Density of Binary Caterpillars

  • Audace A. V. Dossou-OloryEmail author
Original Paper
  • 21 Downloads

Abstract

Given \(d\ge 2\) and two rooted d-ary trees D and T such that D has k leaves, the density \(\gamma (D,T)\) of D in T is the proportion of all k-element subsets of leaves of T that induce a tree isomorphic to D, after contracting all vertices of outdegree 1. In a recent work, it was proved that the limit inferior of this density as the size of T grows to infinity is always zero unless D is the k-leaf binary caterpillar \(F^2_k\) (the binary tree with the property that a path remains upon removal of all the k leaves). Our main theorem in this paper is an exact formula (involving both d and k) for the limit inferior of \(\gamma (F^2_k,T)\) as the size of T tends to infinity.

Keywords

Caterpillars Minimum asymptotic density Leaf-induced subtrees d-ary trees Inducibility Complete d-ary trees Strict d-ary trees 

Mathematics Subject Classification

Primary 05C05 Secondary 05C07 05C30 05C35 

References

  1. 1.
    Czabarka, É., Dossou-Olory, A.A.V., Székely, L.A., Wagner, S.: Inducibility of \(d\)-ary trees (2018). arXiv:1802.03817
  2. 2.
    Czabarka, É., Székely, L.A., Wagner, S.: Inducibility in binary trees and crossings in random tanglegrams. SIAM J. Discrete Math. 31(3), 1732–1750 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dossou-Olory, A.A.V., Wagner, S.: Further results on the inducibility of \(d\)-ary trees. arXiv:1811.11235 (to be submitted)
  4. 4.
    Fischermann, M., Hoffmann, A., Rautenbach, D., Székely, L., Volkmann, L.: Wiener index versus maximum degree in trees. J. Discrete Math. 122(1–3), 127–137 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Székely, L., Wang, H.: On subtrees of trees. Adv. Appl. Math. 34(1), 138–155 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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