A note on coloring sparse random graphs

Coja-Oghlan and Taraz [Amin Coja-Oghlan, Anusch Taraz, Exact and approximative algorithms for coloring G(n,p), Random Structures and Algorithms 24 (3) (2004) 259–278] presented a graph coloring algorithm that has expected linear running time for random graphs with edge probability pp satisfying np≤1.01np≤1.01. In this work, we develop their analysis by exploiting generating function techniques. We show that, in fact, their algorithm colors Gn,pGn,p with the minimal number of colors and has expected linear running time, provided that np≤1.33np≤1.33.