Unique-neighbor Expanders with Better Expansion for Polynomial-sized Sets

Abstract: A $(d_1,d_2)$-biregular bipartite graph $G=(L\cup R,E)$ is called left-$(m,\delta)$ unique-neighbor expander iff each subset $S$ of the left vertices with $|S|\leq m$ has at least $\delta d_1|S|$ unique-neighbors, where unique-neighbors mean vertices with exactly one neighbor in $S$. We can also define right/two-sided expanders similarly. In this paper, we give the following three strongly explicit constructions of unique-neighbor expanders with better unique-neighbor expansion for polynomial-sized sets, while sufficient expansion for linear-sized sets is also preserved: (1) Two-sided $(n^{1/3-\epsilon},1-\epsilon)$ lossless expanders for arbitrary $\epsilon>0$ and aspect ratio. (2) Left-$(\Omega(n),1-\epsilon)$ lossless expanders with right-$(n^{1/3-\epsilon},\delta)$ expansion for some $\delta>0$. (3) Two-sided-$(\Omega(n),\delta)$ unique-neighbor expanders with two-sided-$(n^{\Omega(1)},1/2-\epsilon)$ expansion. The second construction exhibits the first explicit family of one-sided lossless expanders with unique-neighbor expansion for polynomial-sized sets from the other side and constant aspect ratio. The third construction gives two-sided unique-neighbor expanders with additional $(1/2-\epsilon)$ unique-neighbor expansion for two-sided polynomial-sized sets, which approaches the $1/2$ requirement in Lin and Hsieh (arXiv:2203.03581). Our techniques involve tripartite product recently introduced by Hsieh et al (STOC 2024), combined with a generalized existence argument of biregular graph with optimal two-sided unique-neighbor expansion for almost all degrees. We also use a new reduction from large girth/bicycle-freeness to vertex expansion, which might be of independent interest.