Color-avoiding percolation on the Erdős-Rényi random graph

Lyuben Lichev (1) , Bruno Schapira (2)

1 UJM - Université Jean Monnet - Saint-Étienne
2 I2M - Institut de Mathématiques de Marseille

We consider a recently introduced model of color-avoiding 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.

Mots clés: color-avoiding percolation; Erdős-Rényi random graph

Domaines: Probabilités [math.PR]

Lyuben Lichev, Bruno Schapira. Color-avoiding percolation on the Erdős-Rényi random graph. 2022. ⟨hal-03879211⟩