Embedding nearly-spanning bounded degree trees

TL;DR

This paper establishes conditions under which sparse graphs, including random and spectral graphs, contain nearly-spanning bounded degree trees, expanding understanding of embedding trees in sparse structures.

Contribution

It provides new sufficient conditions based on expansion and spectral properties for embedding nearly-spanning bounded degree trees in sparse graphs.

Findings

Random graphs G(n,c/n) contain all such trees with high probability.

Spectral gap conditions ensure embedding of all such trees in (n,D,λ)-graphs.

Derived explicit bounds relating spectral gap to tree embedding capabilities.

Abstract

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most \lambda in their absolute values) has large enough spectral gap D/\lambda as a function of d and \epsilon, then G has a copy of every tree T as above.