Well-mixing vertices and almost expanders

TL;DR

This work studies $D$-regular graphs with a positive fraction of well-mixing vertices and shows they are virtually expanders with no small separators, resolving a question of Pak. It proves that removing at most $5\delta n$ vertices yields a subgraph with strong edge expansion on all small sets, implying large expander substructures and enabling a deterministic polynomial-time algorithm to find long cycles in sparse graphs. It also establishes a cascading improvement: from a positive fraction of well-mixing vertices, almost all vertices become well-mixing after $M+1$ steps, achieving an $O(\tau)$-type mixing time for most vertices. Together, these results connect local random-walk behavior to global expansion, with practical consequences for cycle finding and structural properties in sparse graphs.

Abstract

We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).

Figures (1)

• Figure 1: Relations between well-mixing vertices, expansion, and separators

Theorems & Definitions (25)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4: P02
• Theorem 1.5: P02
• Theorem 1.6
• Theorem 1.7
• Corollary 2.1
• Lemma 2.2
• Theorem 2.3