Packing triangles in bounded degree graphs

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 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 bounded-degree spanning graphs from separable families

Let G be a separable family of graphs. Then for all positive constants and ∆ and for every su ciently large integer n, every sequence G1, . . . , Gt ∈ G of graphs of order n and maximum degree at most ∆ such that e(G1)+· · ·+e(Gt) ≤ (1− ) ( n 2 ) packs into Kn. This improves results of Böttcher, Hladký, Piguet, and Taraz when G is the class of trees and of Messuti, Rödl, and Schacht in the case...

Packing and covering triangles in 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...