First Order Definability of Trees and Sparse Random Graphs
组合数学
2007-05-23 v1
摘要
Let D(G) be the smallest quantifier depth of a first order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the first order descriptive complexity of G. We will show that almost surely D(G)=\Theta(\ln n/\ln\ln n), where G is a random tree of order n or the giant component of a random graph G(n,c/n) with constant c>1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.
引用
@article{arxiv.math/0506288,
title = {First Order Definability of Trees and Sparse Random Graphs},
author = {Tom Bohman and Alan Frieze and Tomasz Luczak and Oleg Pikhurko and Clifford Smyth and Joel Spencer and Oleg Verbitsky},
journal= {arXiv preprint arXiv:math/0506288},
year = {2007}
}
备注
28 pages