# Random graphs with a fixed maximum degree

## TL;DR

This paper analyzes the component structure of random graphs with a fixed maximum degree, identifying a threshold for the emergence of a giant component based on the number of edges.

## Contribution

It establishes a phase transition threshold for the size of the largest component in random graphs with bounded maximum degree.

## Key findings

- Below the threshold, all components are logarithmic in size.
- Above the threshold, a unique giant component emerges.
- The maximum degree constraint influences the phase transition behavior.

## Abstract

We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$ then we establish a threshold value $\mu_\star$ such that if $\mu<\mu_\star$ then w.h.p. the maximum component size is $O(\log n)$. If $\mu>\mu_\star$ then w.h.p. there is a unique giant component of order $n$ and the remaining components have size $O( \log n)$.