Packing triangles in low degree graphs and indifference graphs

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + ε, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of...

Packing triangles in bounded degree graphs

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectiv...

Packing and covering triangles in graphs

Packing and Covering Triangles in Planar Graphs

Tuza conjectured that if a simple graph G does not contain more than k pairwise edgedisjoint triangles, then there exists a set of at most 2k edges that meets all triangles in G. It has been shown that this conjecture is true for planar graphs and the bound is sharp. In this paper, we characterize the set of extremal planar graphs.

Packing Edge-Disjoint Triangles in Given Graphs

Given a graph G, we consider the problem of finding the largest set of edge-disjoint triangles contained in G. We show that even the simpler case of decomposing the edges of a sparse split graph G into edge-disjoint triangles is NP-complete. We show next that the case of a general G can be approximated within a factor of 3 5 in polynomial time, and is NP-hard to approximate within some constant...