On Random Greedy Triangle Packing

Abstract

The behaviour of the random greedy algorithm for constructing a maximal packing of edge-disjoint triangles on $n$ points (a maximal partial triple system) is analysed with particular emphasis on the final number of unused edges. It is shown that this number is at most $n^{7/4+o(1)}$, "halfway" from the previous best-known upper bound $o(n^2)$ to the conjectured value $n^{3/2+o(1)}$. The more general problem of random greedy packing in hypergraphs is also considered.

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

“…Recently Bohman, Frieze and Lubetzky have shown that the random triangle‐removal process terminates a.a.s. with edges improving previous results by Spencer , Rödl and Thoma and Grable .…”

A note on the random greedy triangle-packing algorithm

“…This is in fact the case: It was shown by Spencer [9] and independently by Rödl and Thoma [7] that |E(M)| = o(n 2 ) with high probability 1 . This was extended to |E(M)| ≤ n 11/6+o (1) by Grable in [6], where the author further sketched how similar arguments using more delicate calculations should extend to a bound of n 7/4+o(1) w.h.p. By comparison, it is widely believed that the graph produced by the random greedy triangle-packing process behaves similarly to the Erdős-Rényi random graph with the same edge density, hence the process should end once its number of remaining edges becomes comparable to the number of triangles in the corresponding Erdős-Rényi random graph.…”

“…and the stopping time T j is the minimum of max{j, T } and the smallest index i ≥ j such that Q(i) is not in the critical interval (6). (Note that if Q(j) is not in the critical interval then we have T j = j.)…”

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

“…This problem attracted much attention (see e.g. [15,29,31]), culminating in a result of Bohman, Frieze and Lubetzky [3] where the exponent was finally approximately confirmed. We adapt the triangle removal process so that it does not just produce a partial Steiner triple system, but a k-sparse one.…”