Embedding edge-colored graphs in expanders with roll-back

TL;DR

This paper presents a new method for embedding edge-colored graphs into expander graphs, enabling the detection of complex structures like subdivisions of complete graphs in pseudo-random and finite field vector space graphs.

Contribution

It generalizes existing embedding frameworks and applies them to find structured subgraphs in pseudo-random and finite field distance graphs.

Findings

Edge-colored subdivisions of complete graphs are contained in pseudo-random graphs under certain conditions.

Large subsets of finite field vector spaces contain nearly spanning subdivisions of complete graphs.

The embedding method extends to distance graphs in finite vector spaces.

Abstract

We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Dragani\'c, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently pseudo-random graphs on $n$ vertices contains every edge-colored subdivision of $K_\Delta$, provided that the distance between branch vertices in the subdivision is large enough, the average degree of each graph in the family is at least $(1+o(1))\Delta$, and the number of vertices in the subdivision is at most $(1-o(1))n$. This work is motivated in part by the problem of finding structures in distance graphs defined over finite vector spaces. For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. A distance graph is a graph associated with a non-zero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every distance graph that is a nearly spanning subdivision of a complete graph, provided that the distance between branching vertices in the subdivision is large enough.