The induced-$P_4$-free process

Abstract

We study the random induced-$P_4$-free graph process. Let $e_1,\dots ,e_N$, where $N=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, be a uniformly random ordering of the edges of $K_n$. Starting from the empty graph $G_0$, we add $e_{m+1}$ whenever $G_m+e_{m+1}$ contains no induced $P_4$, and otherwise leave the graph unchanged. We show that the terminal graph is a trivially perfect graph and we describe the structure and distribution of the connected components of the terminal graph $G_N$. Consequently, we derive the limiting values of several natural graph parameters. In particular, the terminal graph $G_N$ has $\Theta (n)$ edges.

32 pages