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September 2016 Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees
Andrea Collevecchio, Abbas Mehrabian, Nick Wormald
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J. Appl. Probab. 53(3): 846-856 (September 2016).


Let r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than tδ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t𝛿, which verifies a conjecture of Cooper and Frieze (2015).


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Andrea Collevecchio. Abbas Mehrabian. Nick Wormald. "Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees." J. Appl. Probab. 53 (3) 846 - 856, September 2016.


Published: September 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1365.05072
MathSciNet: MR3570098

Primary: 05C80
Secondary: 05C05 , 60C05

Keywords: Eggenberger–Pólya urn , longest path , Random Apollonian network , random 𝚤-ary recursive tree

Rights: Copyright © 2016 Applied Probability Trust


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Vol.53 • No. 3 • September 2016
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