Random packing by matroid bases and triangles

Matroid-Based Packing of Arborescences

We provide the directed counterpart of a slight extension of Katoh and Tanigawa’s result [8] on rooted-tree decompositions with matroid constraints. Our result characterizes digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds’ result [1] on packing of spanning arborescences and implies – using a general orientation result of Frank [4] – the a...

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.

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 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.