# The Size of the Giant Joint Component in a Binomial Random Double Graph

## TL;DR

This paper investigates the emergence of a giant joint component in a superposed red and blue binomial random graph, revealing a first-order phase transition characterized by a sudden jump in the size of the largest joint component.

## Contribution

It introduces the concept of joint components in double graphs and identifies critical edge densities where a giant joint component appears with a first-order transition.

## Key findings

- Giant joint component emerges at critical red and blue edge densities.
- The phase transition is first order, with a sudden jump in component size.
- The phenomenon is linked to properties of a bicoloured branching process.

## Abstract

We study the joint components in a random `double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices, which contains both a red and a blue spanning tree. We show that there are critical pairs of red and blue edge densities at which a joint-giant component appears. In contrast to the standard binomial graph model, the phase transition is first order: the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point. We connect this phenomenon to the properties of a certain bicoloured branching process.