Color-avoiding percolation on the Erdős–Rényi random graph

We consider a recently introduced model of color-avoiding percolation (abbreviated CA-percolation) defined as follows. Every edge in a graph G is colored in some of k>=2 colors. Two vertices u and v in G are said to be CA-connected if u and v may be connected using any subset of k-1 colors. CA-connectivity defines an equivalence relation on the vertex set of G whose classes are called CA-components. We study the component structure of a randomly colored Erdős–Rényi random graph of constant average degree. We distinguish three regimes for the size of the largest component: a supercritical regime, a so-called intermediate regime, and a subcritical regime, in which the largest CA-component has respectively linear, logarithmic, and bounded size. Interestingly, in the subcritical regime, the bound is deterministic and given by the number of colors.