Packing Triangles in Weighted Graphs

Chapuy, Guillaume; DeVos, Matt; McDonald, Jessica; Mohar, Bojan; Scheide, Diego

doi:10.1137/100803869

Cited by 16 publications

(87 citation statements)

References 7 publications

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“…• If t overlaps with exactly one triangle ψ ∈ V, we say that t is singly-attached 3 to ψ, and the unique edge in E(t) ∩ E(ψ) is called a base-edge. The vertex anchor(t) := V(t) \ V(ψ) 4 is called the anchoring vertex of t.…”

Section: The Setupmentioning

confidence: 99%

“…For each solution triangle ψ ∈ V, denote the conflict list of ψ by C L(ψ) which contains triangles in T − V whose edges overlap with ψ. We naturally partition C L(ψ) into C L sin (ψ) ∪ C L dou (ψ) ∪ 3 The notion of "attachment" we use here is motivated by the view that triangles in V are the "skeleton" of the graph, and the rest of the graphs are "attachments" hanging around the skeleton. 4 For singleton sets, we abuse notations a bit by interchangeably using e and {e}.…”

Section: Types Of Solution Trianglesmentioning

confidence: 99%

“…If there is no restriction on some dimension, then we put a star ([ * ]) there. 1. type(t) = [3,3]. So the type is either [1,3] or [1,1].…”

Section: Types Of Doubly-attached Trianglesmentioning

confidence: 99%

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Multi-transversals for Triangles and the Tuza's Conjecture

Chalermsook¹,

Khuller²,

Sukprasert³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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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) ≤ 3ν(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) ≤ 2ν(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| ≤ 2kν(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.Clearly, Conjecture 1 is a consequence of the Tuza's conjecture and it implies the standard fractional version of Tuza's conjecture that was resolved by Krivelevich. Krivelevich's proof is based on induction and only implies that τ * k (G) ≤ 2ν(G) for some (very large) k ∈ N. 2 In this paper, we resolve the above question in the affirmative.Theorem 2. For all integer k ≥ 2 and any graph G, τ * k (G) ≤ 2ν(G). Moreover, we can efficiently find the triangle packing solution and the k-multi-transversal that together achieve this bound.The proof of this theorem is inspired by the local search technique. Based on an optimal packing solution, we define a collection of edges in the covering solution via the local view of the packing solution. Our proofs are constructive in nature. Starting with any packing solution, we can either find a multi-transversal or improve the current packing solution. Because the packing solution can be improved at most O(n 3 ) times, hence we get an algorithm to find the triangle packing solution and the k-multi-transversal that together achieve this bound.We believe that the structural insights from this work would be useful in attacking the Tuza's conjecture in the future.

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Section: The Setupmentioning

confidence: 99%

Section: Types Of Solution Trianglesmentioning

confidence: 99%

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Multi-transversals for Triangles and the Tuza's Conjecture

Chalermsook¹,

Khuller²,

Sukprasert³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The best general upper bound on τ (G) in terms of ν(G) is due to Haxell [12], who showed that τ (G) ≤ 2.87ν(G) for all graphs G. Tuza's Conjecture has been studied by many authors, who proved the conjecture for special classes of graphs [14,18,23,25,26] or studied various fractional relaxations of the conjecture [6,13,15,18].…”

mentioning

confidence: 99%

“…A similar idea, restricted to k = 1, appears in [15] and [6], where H is taken to be a triangle-free Ramsey graph with small independence number, and graphs of the form I 1 ∨ H are used as sharpness examples for upper bounds on τ (G).…”

mentioning

confidence: 99%

Maximal k-edge-colorable subgraphs, Vizing’s Theorem, and Tuza’s Conjecture

Puleo

2017

Discrete Mathematics

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Abstract. We prove that if M is a maximal k-edge-colorable subgraph of a multigraph G and if(When G is a simple graph, the set F is just the set of vertices having degree less than k in M .) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.

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