Cover time of a random graph with a degree sequence II: Allowing vertices of degree two

TL;DR

This paper extends the understanding of cover times in random graphs by allowing vertices of degree two, providing asymptotic results for the 2-core of the giant component in Erdős-Rényi graphs.

Contribution

It generalizes previous results by relaxing degree constraints, specifically allowing degree-two vertices, and applies findings to the 2-core of the giant component.

Findings

Asymptotic cover time for graphs with degree ≥ 2

Results for the 2-core of the giant component in Erdős-Rényi graphs

Extension of previous models with degree constraints

Abstract

We study the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\mathbf{d}=(d_i)_{i=1}^n$. In a previous work, the asymptotic cover time was obtained under a number of assumptions on $\mathbf{d}$, the most significant being that $d_i\geq 3$ for all $i$. Here we replace this assumption by $d_i\geq 2$. As a corollary, we establish the asymptotic cover time for the 2-core of the emerging giant component of $\mathcal{G}(n,p)$.