Making Triangles Colorful

Colorful Triangle Counting and a MapReduce Implementation

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if p ≥ max ( log n t , log n √ t ), where n, t, ∆ denote the number of vertices in G, the number of triangles in G, the maximum number of triangles an edge of G is cont...

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation m◦n ≡ m+n−2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorp...

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.

Making Octants Colorful and Related Covering Decomposition Problems

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R can be colored with k colors so that every translate of the negative octant containing at least k points contains at least one of each color. The best previously known bound was doubly...

The Ill-Balanced Triangles of the Islamic States in Crisis