Large triangle packings and Tuza’s conjecture in sparse random graphs

Abstract

The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

Tuza's Conjecture for random graphs

“…We recently heard from Patrick Bennett that he, Ryan Cushman and Andrzej Dudek [2] have also closed the gap in [3], using an approach similar to that of the earlier paper (and different from what we do here).…”

“…For d > log 3+ n Theorem 1.7 was proved (somewhat implicitly) in[4], and, as observed in[3], direct application of Pippenger's Theorem improves this to d log n, where w.h.p. each edge of G is in (1 + o(1))d triangles.…”

“…In [3] this was shown to be true if p < c 1 n −1/2 or p > c 2 n −1/2 , with c 1 ≈ 0.48 and c 2 ≈ 4.25. (They work with G n,m , but, as usual, this is about the same as G n,p with p = m/ n 2 and we will stick to the binomial version.)…”

“…For the rest of this paper we use G for G n,p , and set m = n 2 p (= E|G|) and d = (n − 2)p 2 (the expected number of triangles on a given edge of G). Of course for the proof of Theorem 1.2 we could confine ourselves to d in the range not covered by [3], but we will give arguments for the full range, in the process strengthening the earlier results.…”

“…More precisely, is it true that for any p = p(n) and G n,p the usual binomial (or "Erdős-Rényi") random graph, w.h.p. 1G n,p satisfies Tuza's Conjecture?In [3] this was shown to be true if p < c 1 n −1/2 or p > c 2 n −1/2 , with c 1 ≈ 0.48 and c 2 ≈ 4.25. (They work with G n,m , but, as usual, this is about the same as G n,p with p = m/ n 2 and we will stick to the binomial version.)…”