Optimal packing of similar triangles

Similar Triangles, Another Trace of the Golden Ratio

In this paper we investigate similar triangles which are not congruent but have two pair congruent sides. We show that greatest lower bound of values for similarity ratio of such triangles is golden ratio. For right triangles, we prove that the supremum of values for similarity ratio is the square root of the golden ratio.

Online Packing of Equilateral Triangles

We investigate the online triangle packing problem in which a sequence of equilateral triangles with different sizes appear in an online, sequential manner. The goal is to place these triangles into a minimum number of squares of unit size. We provide upper and lower bounds for the competitive ratio of online algorithms. In particular, we introduce an algorithm which achieves a competitive rati...

Packing Triangles in Regular Tournaments

We prove that a regular tournament with n vertices has more than n 2 11.5 (1 − o(1)) pairwise arc-disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than n 2 8 , so these tournaments do not have n 8 pairwise arc-disjoint triangles. These improve upon the best known bounds for this problem.

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...