On the cover time of dense graphs

TL;DR

This paper provides asymptotic estimates and approximations for the cover time of dense graphs with high minimum degree, depending on conductance and mixing time properties, advancing understanding of random walk behaviors.

Contribution

It introduces new asymptotic formulas and approximation methods for cover time in dense graphs based on conductance and mixing time conditions.

Findings

Asymptotic expression for cover time when conductance is large.

Deterministic estimate of cover time via graph decomposition.

Approximate cover time with a (2+o(1))-factor in general cases.

Abstract

We consider arbitrary graphs $G$ with $n$ vertices and minimum degree at least $\delta n$ where $\delta>0$ is constant. If the conductance of $G$ is sufficiently large then we obtain an asymptotic expression for the cover time $C_G$ of $G$ as the solution to an explicit transcendental equation. Failing this, if the mixing time of a random walk on $G$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense sub-graphs with high conductance. Failing this we give a deterministic asymptotic (2+o(1))-approximation of $C_G$.