Packing and Covering Triangles in Dense Random Graphs

Abstract

Given a simple graph $G=(V,E)$, a subset of E is called a triangle cover if it intersects each triangle of G. Let $\nu _t(G)$ and $\tau _t(G)$ denote the maximum number of pairwise edge-disjoint triangles in G and the minimum cardinality of a triangle cover of G, respectively. Tuza [25] conjectured in 1981 that $\tau _t(G)/\nu _t(G)\le 2$ holds for every graph G. In this paper, we consider Tuza’s Conjecture on dense random graphs. We prove that under $\mathcal {G}(n,p)$ model with $p=\varOmega (1)$, for any $0<\epsilon <1$, $\tau _t(G)\le 1.5(1+\epsilon )\nu _t(G)$ holds with high probability, and under $\mathcal {G}(n,m)$ model with $m=\varOmega (n^2)$, for any $0<\epsilon <1$, $\tau _t(G)\le 1.5(1+\epsilon )\nu _t(G)$ holds with high probability. In some sense, on dense random graphs, these conclusions verify Tuza’s Conjecture.

This research is supported part by National Natural Science Foundation of China under Grant No. 11901605, and by the disciplinary funding of Central University of Finance and Economics.

Appendix: A List of Mathematical Symbols

$\mathcal {G}(n,p)$	Given $0\le p\le 1$, $\mathbf {Pr}(\{v_i,v_j\}\in G)=p$ for all $v_i,v_j$
	With these probabilities mutually independent
$\mathcal {G}(n,m)$	Given $0\le m\le n(n-1)/2$, let G be defined by
	Randomly picking m edges from all $v_i, v_j$ pairs
$\tau _t(G)$	The minimum cardinality of a triangle cover in G
$\nu _t(G)$	The maximum cardinality of a triangle packing in G
$\tau ^{*}_t(G)$	The minimum cardinality of a fractional triangle cover in G
$\nu ^{*}_t(G)$	The maximum cardinality of a fractional triangle packing in G
b(G)	The maximum number of edges of sub-bipartite in G
$\delta (G)$	The minimum degree of graph G
$f(n)=O(g(n))$	$\exists ~c>0, n_{0}\in \mathbb {N}_{+},\forall n\ge n_{0}, 0\le f(n)\le cg(n)$
$f(n)=\varOmega (g(n))$	$\exists ~c>0, n_{0}\in \mathbb {N}_{+},\forall n\ge n_{0}, 0\le cg(n)\le f(n)$
$f(n)=\varTheta (g(n))$	$\exists ~c_{1}>0, c_{2}>0, n_{0}\in \mathbb {N}_{+},\forall n\ge n_{0}$,$0\le c_{1}g(n)\le f(n)\le c_{2}g(n)$
$f(n)=o(g(n))$	$\forall ~c>0, \exists ~n_{0}\in \mathbb {N}_{+},\forall n\ge n_{0}, 0\le f(n)< cg(n)$
$f(n)=\omega (g(n))$	$\forall ~c>0, \exists ~n_{0}\in \mathbb {N}_{+},\forall n\ge n_{0}, 0\le cg(n)< f(n)$

Union Bound Inequality:

For any finite or countably infinite sequence of events $E_1,E_2,\dots$, then

$$\mathbf {Pr}\left[ ~\bigcup _{i\ge 1} E_i~\right] \le \sum _{i\ge 1} \mathbf {Pr}(E_i).$$

Chernoff’s Inequalities:

Let $X_1,X_2,\dots ,X_n$ be mutually independent 0–1 random variables with $\mathbf {Pr}[X_i = 1] = p_i$. Let $X = \sum _{i=1}^n X_i$ and $\mu = \mathbf{E}[X]$. For $0 < \epsilon \le 1$, then the following bounds hold:

$$ \mathbf {Pr}[X \ge (1+\epsilon )\mu ] \le e^{-\epsilon ^2 \mu /3},~~~~\mathbf {Pr}[X \le (1-\epsilon )\mu ] \le e^{-\epsilon ^2 \mu /2}. $$

Chebyshev’s Inequality:

For any $a > 0$,

$$\mathbf {Pr}[|X-\mathbf{E}[X]|\ge a] \le \frac{\mathbf {Var}[X]}{a^2}.$$