Abstract
We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and $m$-ary search trees, preferential attachment trees, and others. The limiting constant is computed, analytically or numerically, for several examples. The method is based on Crump–Mode–Jagers branching processes.
Highlights
We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and m-ary search trees, preferential attachment trees, and others
The independence number i.e., the maximum size of an independent set of nodes, is a quantity that has been studied for various models of random trees
In the present paper we consider rooted trees that can be constructed as family trees of a Crump–Mode–Jagers branching process stopped at a suitable stopping time; this includes, for example, random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and m-ary search trees; see Section 2.1 and [8] for details, and the examples in Sections 4–8 below
Summary
The independence number i.e., the maximum size of an independent set of nodes, is a quantity that has been studied for various models of random trees (and other random graphs, not considered here). Theorem 3.1, gives a strong law of large numbers for I(T ); more precisely, it shows convergence almost surely (a.s.) of I(Tn)/|Tn|, the fraction of nodes that belong to a maximum independent set, for a sequence Tn of random trees. The expectation was found already by Meir and Moon [12, 15] Both Dadedzi [4] and Fuchs et al [6] prove (independently, and with different methods) the weak version For trees, several other quantities are determined by the independence number by linear relations, and our results immediately transfer to these quantities See e.g. [6] for further examples
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