Rolling backwards can move you forward: on embedding problems in sparse expanders

TL;DR

This paper introduces a novel embedding method with a rollback feature, leading to significant results in graph theory, including linear size-Ramsey numbers, an online path-finding algorithm, and bounds on topological minors in expanders.

Contribution

It develops a new embedding technique with rollback capability and applies it to solve open problems in graph embeddings, Ramsey theory, and spectral graph properties.

Findings

Size-Ramsey number of subdivisions is linear in vertices.

Deterministic online algorithm for vertex-disjoint paths in expanders.

Bounds on spectral ratio imply existence of large topological minors.

Abstract

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. We use this method to obtain the following results. -We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak (2002). -We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo (2007). -We show that relatively weak bounds on the spectral ratio of $d$-regular graphs force the existence of a topological minor of $K_t$ where $t=(1-o(1))d$. We also exhibit a construction which shows that the theoretical maximum $t=d+1$ cannot be attained even if $\lambda=O(\sqrt{d})$. This answers a question of Fountoulakis, K\"uhn and Osthus (2009).