Core Concepts
The authors determine the order of magnitude of the clique chromatic number of sparse random graphs, resolving open problems and discovering new phenomena.
Abstract
The paper analyzes the clique chromatic number of random graphs, focusing on edge-probabilities in specific ranges. The authors address challenges related to high-degree vertices impacting maximal cliques. By utilizing a union bound argument and Janson's inequality, they determine asymptotics and reveal surprising results contradicting earlier predictions.
Stats
The typical value of χc(Gn,p) is determined up to constant factors for various edge-probabilities.
For most p in the range n−1 ≪ p ≤ n−2/5−ε, χc(Gn,p) = Θ(np log(np)).
For most p in the range n−2/5+ε ≤ p ≤ n−1/3−ε or n−1/3+ε ≤ p ≤ n−ε, χc(Gn,p) = ˜Θ(1/p).
In the sparse range n−o(1) ≤ p ≪ 1, χc(Gn,p) = 1/2 + o(1) log(n)/p.
For very small edge-probabilities (log n)^ωn^(-2/5) ≤ p ≪ n^(-1/3), χc(Gn,p) = 5/2 + o(1) log(n^(2/5)p)/p.
Quotes
"The clique chromatic number analysis reveals surprising results that challenge previous predictions."
"The study addresses complexities arising from high-degree vertices impacting maximal cliques."