Color-avoiding percolation of random graphs: between the subcritical and the intermediate regime

TL;DR

This paper investigates the phase transition in color-avoiding percolation of random graphs, focusing on the transition between the intermediate and subcritical regimes where the largest CA-component size shifts from logarithmic to bounded.

Contribution

It characterizes the phase transition between the intermediate and subcritical regimes in the size of the largest CA-component in randomly colored graphs.

Findings

Identifies the critical point of the phase transition.

Describes the behavior of CA-components near the transition.

Provides theoretical insights into the size scaling of CA-components.

Abstract

Fix a graph $G$ in which every edge is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ are CA-connected if $u$ and $v$ may be connected using any subset of $k - 1$ colors. CA-connectivity is an equivalence relation dividing the vertex set into classes called CA-components. In two recent papers, R\'ath, Varga, Fekete, and Molontay, and Lichev and Schapira studied the size of the largest CA-component in a randomly colored random graph. The second of these works distinguished and studied three regimes (supercritical, intermediate, and subcritical) in which the largest CA-component has respectively linear, logarithmic, and bounded size. In this short note, we describe the phase transition between the intermediate and the subcritical regime.