# Covering Triangles in Edge-Weighted Graphs

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## Abstract

Let *G* = (*V*, *E*) be a simple graph and \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\) assign each edge *e* ∈ *E* a positive integer weight *w*(*e*). A subset of *E* that intersects every triangle of *G* is called a triangle cover of (*G*, **w**), and its weight is the total weight of its edges. A collection of triangles in *G* (repetition allowed) is called a triangle packing of (*G*, **w**) if each edge *e* ∈ *E* appears in at most *w*(*e*) members of the collection. Let *τ*_{ t }(*G*, **w**) and *ν*_{ t }(*G*, **w**) denote the minimum weight of a triangle cover and the maximum cardinality of a triangle packing of (*G*, **w**), respectively. Generalizing Tuza’s conjecture for unit weight, Chapuy et al. conjectured that *τ*_{ t }(*G*, **w**)/*ν*_{ t }(*G*, **w**) ≤ 2 holds for every simple graph *G* and every \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\). In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding triangle covers of small weights. These algorithms imply new sufficient conditions for the conjecture of Chapuy et al. More precisely, given (*G*, **w**), suppose that all edges of *G* are covered by the set \({\mathscr {T}}_{G}\) consisting of edge sets of triangles in *G*. Let \(|E|_{w}={\sum }_{e\in E}w(e)\) and \(|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)\) denote the weighted numbers of edges and triangles in (*G*, **w**), respectively. We show that a triangle cover of (*G*, **w**) of weight at most 2*ν*_{ t }(*G*, **w**) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) \(\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}\), (ii) \(\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}\), (iii) \(|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2\).

## Keywords

Triangle cover Triangle packing Linear 3-uniform hypergraphs Combinatorial algorithms## Notes

### Acknowledgments

The authors are indebted to Dr. Gregory J. Puleo, Dr. Zbigniew Lonc, and two anonymous referees for their invaluable comments and suggestions.

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