The $k$-representation number of the random graph

Abstract

The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In particular, for $k=1$ this notion is equivalent to the clique number of a graph $G$. Extending results of Frieze and Reed, and Eaton and Grable, we study the $k$-representation number of $G(n,1/2)$. As a tool, we will prove a sharp concentration result counting the number of induced subgraphs of $G(n,1/2)$ with density $(\frac{1}{2}+\alpha )$. In Lemma 3.7, we will show that the number of such subgraphs is close to its expected value with probability $1-\exp (-n^C)$.