Embedding edge-colored graphs in expanders with roll-back
Ben Lund, Chuandong Xu
TL;DR
The paper develops a colorful roll-back framework to embed large edge-colored graphs into families of expander/jumbled graphs, extending prior roll-back methods to handle multiple colors and complex structures. Central technical tools include Forest Extension and Path Connection lemmas, enabling systematic extension of good embeddings and construction of edge-colored subdivisions of complete graphs. The authors prove that sufficiently pseudo-random graph families on $n$ vertices contain every $[t]$-edge-colored subdivision of $K_D$ with long spacing between branch vertices, and they translate these results to distance graphs in finite vector spaces, showing large subsets contain nearly spanning distance-graph subdivisions. This work advances imbedability results in pseudorandom environments and has implications for geometric incidence problems in finite fields.
Abstract
We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Draganić, 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 $K_Δ$, 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 $(1+o(1))Δ$, 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.