# Covering Triangles in Edge-Weighted Graphs

Covering Triangles in Edge-Weighted Graphs Let G = (V, E) be a simple graph and w ∈ ℤ > 0 E $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ 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 w ∈ ℤ > 0 E $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ . 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 T G ${\mathscr {T}}_{G}$ consisting of edge sets of triangles in G. Let | E | w = ∑ e ∈ E w ( e ) $|E|_{w}={\sum }_{e\in E}w(e)$ and | T G | w = ∑ { e , f , g } ∈ T G w ( e ) w ( f ) w ( g ) $|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)$ 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 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) ν t ( G , w ) / | T G | w ≥ 1 3 $\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}$ , (ii) ν t ( G , w ) / | E | w ≥ 1 4 $\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}$ , (iii) | E | w / | T G | w ≥ 2 $|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

# Covering Triangles in Edge-Weighted Graphs

, Volume 62 (6) – Mar 23, 2018
28 pages

/lp/springer_journal/covering-triangles-in-edge-weighted-graphs-wzJsxJ2wCV
Publisher
Springer US
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-018-9860-7
Publisher site
See Article on Publisher Site

### Abstract

Let G = (V, E) be a simple graph and w ∈ ℤ > 0 E $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ 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 w ∈ ℤ > 0 E $\mathbf {w}\in \mathbb {Z}^{E}_{>0}$ . 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 T G ${\mathscr {T}}_{G}$ consisting of edge sets of triangles in G. Let | E | w = ∑ e ∈ E w ( e ) $|E|_{w}={\sum }_{e\in E}w(e)$ and | T G | w = ∑ { e , f , g } ∈ T G w ( e ) w ( f ) w ( g ) $|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)$ 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 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) ν t ( G , w ) / | T G | w ≥ 1 3 $\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}$ , (ii) ν t ( G , w ) / | E | w ≥ 1 4 $\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}$ , (iii) | E | w / | T G | w ≥ 2 $|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2$ .

### Journal

Theory of Computing SystemsSpringer Journals

Published: Mar 23, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations