[Paper Review] Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Xujin Chen, Zhuo Diao+2 more|arXiv (Cornell University)|May 6, 2016

Advanced Graph Theory Research13 references3 citations

TL;DR

This paper proposes polynomial-time combinatorial algorithms for finding small triangle covers in graphs, using a hypergraph approach to establish new sufficient conditions for Tuza's conjecture. It proves that τₜ(G) ≤ 2νₜ(G) holds if νₜ(G)/|𝒯_G| ≥ 1/3, νₜ(G)/|E| ≥ 1/4, or |E|/|𝒯_G| ≥ 2, thereby verifying the conjecture under these structural constraints.

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 $ν_t(G)$ and $τ_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $τ_t(G)/ν_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2ν_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $ν_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $ν_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithms

Motivation & Objective

• To establish new sufficient conditions under which Tuza’s conjecture τₜ(G) ≤ 2νₜ(G) holds for all graphs.
• To design efficient, polynomial-time combinatorial algorithms for computing small triangle covers in graphs.
• To bridge graph triangle packing and covering with linear 3-uniform hypergraph theory, particularly acyclicity and transversal bounds.
• To investigate the structural thresholds where Tuza’s conjecture is tight or near-tight, focusing on extremal graphs.
• To generalize results from graphs to linear 3-uniform hypergraphs, identifying cases where τ(ℋ) ≤ 2ν(ℋ) holds computationally.

Proposed method

• Model the triangle hypergraph ℋ_G = (E, 𝒯_G) as a linear 3-uniform hypergraph, where edges represent triangles.
• Use hypergraph acyclicity as a key structural property: removing at most νₜ(G) edges makes ℋ_G acyclic, ensuring τₜ(G') = νₜ(G') for the remaining graph G'.
• Apply a polynomial-time algorithm to compute a minimum triangle cover in the acyclic subgraph G', then combine it with the removed edges to form a cover of size ≤ 2νₜ(G).
• Leverage known results on linear 3-uniform hypergraphs, particularly that a transversal of size ≤ 2ν(ℋ) can be found in polynomial time if ν(ℋ)/|ℰ| ≥ 1/3 or |𝒱|/|ℰ| ≥ 2.
• Use the hypergraph framework to prove that τₜ(G) ≤ 2νₜ(G) holds under three sufficient conditions: (i) νₜ(G)/|𝒯_G| ≥ 1/3, (ii) νₜ(G)/|E| ≥ 1/4, (iii) |E|/|𝒯_G| ≥ 2.
• Analyze extremal graphs satisfying τₜ(G)/νₜ(G) = 2 and show they satisfy the reverse inequalities: νₜ(G)/|𝒯_G| ≤ 1/3, νₜ(G)/|E| ≤ 1/4, |E|/|𝒯_G| ≤ 2.

Experimental results

Research questions

• RQ1 Under what structural conditions on a graph G does Tuza’s conjecture τₜ(G) ≤ 2νₜ(G) hold?
• RQ2 Can polynomial-time algorithms be designed to compute triangle covers of size at most 2νₜ(G) under specific density or packing constraints?
• RQ3 How do the ratios νₜ(G)/|𝒯_G|, νₜ(G)/|E|, and |E|/|𝒯_G| relate to the validity of Tuza’s conjecture?
• RQ4 What is the role of hypergraph acyclicity in enabling efficient computation of small triangle covers?
• RQ5 Can the results on graphs be generalized to linear 3-uniform hypergraphs, and what are the limitations of such generalization?

Key findings

• A triangle cover of size at most 2νₜ(G) can be computed in polynomial time if νₜ(G)/|𝒯_G| ≥ 1/3, νₜ(G)/|E| ≥ 1/4, or |E|/|𝒯_G| ≥ 2.
• The sufficient condition νₜ(G)/|𝒯_G| ≥ 1/3 implies that at least one-third of all triangles are edge-disjoint, enabling efficient cover construction.
• The condition |E|/|𝒯_G| ≥ 2 means that the average number of edges per triangle is at least 2, indicating high triangle density and structural sparsity.
• Extremal graphs achieving τₜ(G)/νₜ(G) = 2 satisfy the reverse inequalities: νₜ(G)/|𝒯_G| ≤ 1/3, νₜ(G)/|E| ≤ 1/4, and |E|/|𝒯_G| ≤ 2.
• The Fano plane provides a counterexample showing that τ(ℋ) ≤ 2ν(ℋ) does not hold for all linear 3-uniform hypergraphs, limiting the generalization of the result.
• For Erdős–Rényi random graphs G(n,p), Tuza’s conjecture holds with high probability when p > √3/2, suggesting a threshold behavior near this value.