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arXiv · 2608.18648

The induced-$P_4$-free process

Hongyi Lou·Xinzhe Song·Guiying Yan
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Abstract

We study the random induced-$P_4$-free graph process. Let $e_1,\ldots,e_N$, where $N=\binom{n}{2}$, 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.

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Hongyi Lou, Xinzhe Song, Guiying Yan. 2026-08-19. The induced-$P_4$-free process. https://arxiv.org/abs/2608.18648

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