A note on the width of sparse random graphs

TL;DR

The width of a supercritical random graph according to some commonly studied width measures is considered and short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra are given on the rankand tree-width of the random graph G(n, p) when p = 1+ǫ n for ǫ > 0 constant.

Abstract

In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+ε}{n}$ for $ε> 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $ε$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $ε= o(1)$ and $ε^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $ε$.

Figures (4)

• Figure 1: The local complementation $G*v$ of a graph $G$ at a vertex $v$.
• Figure 2: Whp for every partition $(V_1,V_2)$ of the vertex set of the tree $T \subseteq G(n,p_1)$, which we draw in red, with few crossing edges in $T$, there are many crossing edges in $G(n,p_2)$, drawn in blue.
• Figure 3: A graph $G$, its 2-core $C$, and its kernel $K$.
• Figure 4: The effect of contracting an edge $m$ in a configuration on a set of cells $\mathcal{V}$. The cells $V_2$ and $V_3$ containing the two half-edges $e$ and $f$ matched by $m$ are replaced by a single cell $V_{2,3}$.

Theorems & Definitions (34)

• Theorem 1.1: LeeLeeOum
• Theorem 1.2: Serra
• Theorem 1.3: Bejamini2014
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: See Harvey
• Lemma 2.2
• proof
• Lemma 2.3: LeeLeeOum