A note on the random triadic process

Abstract

For a fixed integer $r⩾3$, let $H_r(n,p)$ be a random r-uniform hypergraph on the vertex set $\left[n\right]$, where each r-set is an edge randomly and independently with probability p. The random r-generalized triadic process starts with a complete bipartite graph $K_{r-2,n-r+2}$ on the same vertex set, chooses two distinct vertices x and y uniformly at random and iteratively adds $\{x,y\}$ as an edge if there is a subset Z with size $r-2$, denoted as $Z=\{z_1,\cdots ,z_{r-2}\}$, such that $\{x,z_i\}$ and $\{y,z_i\}$ for $1⩽i⩽r-2$ are already edges in the graph and $\{x,y,z_1,\cdots ,z_{r-2}\}$ is an edge in $H_r(n,p)$. The random triadic process is an abbreviation for the random 3-generalized triadic process. Korándi et al. proved a sharp threshold probability for the propagation of the random triadic process, that is, if $p=c{n}^{-\frac{1}{2}}$ for some positive constant c, with high probability, the triadic process reaches the complete graph when $c>\frac{1}{2}$ and stops at $O\left(n^{\frac{3}{2}}\right)$ edges when $c<\frac{1}{2}$. In this note, we consider the final size of the random r-generalized triadic process when $p=o(n^{-\frac{1}{2}}{\log }^{\alpha (3-r)}n)$ with a constant $\alpha >\frac{1}{2}$. We show that the generated graph of the process essentially behaves like $G(n,p)$. The final number of added edges in the process, with high probability, equals $\frac{1}{2}{n}^{2}p(1+o(1\left)\right)$ provided that $p=\omega \left(n^{-2}\right)$. The results partially complement the ones on the case of $r=3$.