Tree decompositions of graphs without large bipartite holes

TL;DR

This paper proves that graphs without large bipartite holes can be approximately decomposed into bounded degree trees, extending previous results and introducing new methods in the context of Ramsey-Turán theory and random perturbations.

Contribution

It establishes a new approximate decomposition result for almost va n-regular graphs without large bipartite holes, generalizing prior work and connecting to random graph models.

Findings

Graphs without large bipartite holes admit approximate decompositions into bounded degree trees.

The result is sharp; large bipartite holes prevent such decompositions.

Randomly perturbed graphs typically have such decompositions with high probability.

Abstract

A recent result of Condon, Kim, K\"{u}hn and Osthus implies that for any $r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost $\alpha n$-regular graph $G$ with any $\alpha>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into $n$-vertex trees. Moreover, this implies that for any $\alpha>0$ and an $n$-vertex almost $\alpha n$-regular graph $G$, with high probability, the randomly perturbed graph $G\cup \mathbf{G}(n,O(\frac{1}{n}))$ has an approximate decomposition into all collections of bounded degree trees of size at most $(1-o(1))n$ simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey-Tur\'an theory and the randomly perturbed graph model.