Random $K_k$-removal algorithm

TL;DR

This paper analyzes the random $K_k$-removal algorithm, determining the expected evolution of key parameters and the final size of the resulting hypergraph for $k \\geq 4$, extending previous results for $k=3$.

Contribution

It provides the exact expected trajectories of key parameters and bounds the final size for the random $K_k$-removal process when $k \\geq 4$, advancing understanding of this graph evolution.

Findings

Expected trajectories of key parameters are derived.

Final size of the hypergraph is bounded by $n^{2-1/(k(k-1)-2)+o(1)}$ for $k \\geq 4$.

The bound is shown to be a natural barrier.

Abstract

One interesting question is how a graph develops from some constrained random graph process, which is a fundamental mechanism in the formation and evolution of dynamic networks. The problem here is referred to the random $K_k$-removal algorithm. For a fixed integer $k\geqslant 3$, it starts with a complete graph on $n\rightarrow\infty$ vertices and iteratively removes the edges of an uniformly chosen $K_k$. This algorithm terminates once no $K_k$s remain and at the same time it generates one linear $k$-uniform hypergraph. For $k=3$, it was shown that the size in the final graph is $n^{3/2+o(1)}$. Less results are on the cases when $k\geqslant 4$. In this paper, we prove that the exact expected trajectories of various key parameters in the algorithm to some iteration such that the final size in the algorithm is at most $n^{2-1/(k(k-1)-2)+o(1)}$ for $k\geqslant 4$. We also show the bound is a natural barrier.