Sapozhenko’s graph lemma

In this note we discuss a lemma from [3]. Before stating the lemma, we need to establish some notation. This is done in Section 1. We then state Sapozhenko's result in Section 2; that section also previews the rest of the note. 1 Setup Throughout, Σ is a d-regular bipartite graph with bipartition V = V (Σ) = X ∪ Y. For u, v ∈ V and A, B ⊆ V we write u ∼ v if there is an edge in Σ joining u and v, ∇(A) for the set of edges having one end in A and ∇(A, B) for the set of edges having one end in each of A, B. Set N (u) = {w ∈ V : w ∼ u} (N (u) is the neighbourhood of u), N (A) = ∪ w∈A N (w), N B (u) = {w ∈ B : w ∼ u}, N B (A) = ∪ w∈A N B (w), d(u) = |N (u)| and d B (u) = |N B (u)|. Write ρ(u, v) for the length of the shortest u-v path in Σ, and set ρ(u, A) = min w∈A {ρ(u, w)}