DeepAI

# Multi-transversals for Triangles and the Tuza's Conjecture

In this paper, we study a primal and dual relationship about triangles: For any graph G, let ν(G) be the maximum number of edge-disjoint triangles in G, and τ(G) be the minimum subset F of edges such that G ∖ F is triangle-free. It is easy to see that ν(G) ≤τ(G) ≤ 3 ν(G), and in fact, this rather obvious inequality holds for a much more general primal-dual relation between k-hyper matching and covering in hypergraphs. Tuza conjectured in 1981 that τ(G) ≤ 2 ν(G), and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every k ≥ 2, there exist a (multi)-set F ⊆ E(G): |F| ≤ 2k ν(G) such that each triangle in G overlaps at least k elements in F. Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in F based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.

• 14 publications
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05/20/2021

### Tuza's Conjecture for Threshold Graphs

Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjo...
11/14/2022

### On Tuza's conjecture in co-chain graphs

In 1981, Tuza conjectured that the cardinality of a minimum set of edges...
02/18/2020

### On Tuza's conjecture for triangulations and graphs with small treewidth

Tuza (1981) conjectured that the size τ(G) of a minimum set of edges tha...
12/20/2018

### Strengthening the Murty-Simon conjecture on diameter 2 critical graphs

A graph is diameter-2-critical if its diameter is 2 but the removal of a...
11/16/2017

### Packing nearly optimal Ramsey R(3,t) graphs

In 1995 Kim famously proved the Ramsey bound R(3,t) > c t^2/ t by constr...
05/06/2016

### Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph G=(V,E), a subset of E is called a triangle cover i...
11/01/2019

### Computing Circle Packing Representations of Planar Graphs

The Circle Packing Theorem states that every planar graph can be represe...