Transversals in trees
Combinatorics
2013-08-20 v1
Abstract
A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree.
Keywords
Cite
@article{arxiv.0705.1806,
title = {Transversals in trees},
author = {Victor Campos and Vasek Chvatal and Luc Devroye and Perouz Taslakian},
journal= {arXiv preprint arXiv:0705.1806},
year = {2013}
}