Large expanders in high genus unicellular maps

Abstract

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.

Finding Large Expanders in Graphs: from Topological Minors to Induced Subgraphs

“…It is easy to check that a ′ " ca and b ′ " cb together imply min{a ′ , b ′ } " c min{a, b} and therefore, by ( 9), ( 10) and (11):…”

“…This result can be helpful to study some models of graphs where the topology is an important feature, such as combinatorial maps. In particular, Theorem 1 is a key tool in [11], which studies maps in the hyperbolic high genus regime, and the presence of large expander subgraphs in them. Our result is important for this problem as it then allows to use and study the classical "core-kernel decomposition", which produces topological minors of maps.…”