A note on sublinear separators and expansion

Abstract

For a hereditary class G of graphs, let sG(n) be the minimum function such that each n-vertex graph in G has a balanced separator of order at most sG(n), and let ∇G(r) be the minimum function bounding the expansion of G, in the sense of bounded expansion theory of Nešetřil and Ossona de Mendez. The results of Plotkin et al. (1994) and Esperet and Raymond (2018) imply that if sG(n)=Θ(n1−ε) for some ε>0, then ∇G(r)=Ω(r12ε−1∕polylog r) and ∇G(r)=O(r1ε−1polylog r). Answering a question of Esperet and Raymond, we show that neither of the exponents can be substantially improved.