Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

Coloring planar homothets and three-dimensional hypergraphs

We prove that every finite set of homothetic copies of a given convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any k≥2, every three-dimensional hypergraph can be colored with 6(k-1) colors so that every hyperedge e contains min{SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Ordered coloring grids and related graphs

We investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) is related to conflict-free coloring and other traditional coloring problems. Such coloring problems can model (among others) efficient frequency assignments in cellular networks. Our main technical results improve upper and lower bounds for the ordered chromatic number of grids and related graphs. To the best of our knowledge, this is the first attempt to calculate exactly the ordered chromatic number of these graph families

Evidence Accumulation Clustering based on the K-Means Algorithm

The idea of evidence accumulation for the combination of multiple clusterings was recently proposed [7]. Taking the K-means as the basic algorithm for the decomposition of data into a large number, k, of compact clusters, evidence on pattern association is accumulated, by a voting mechanism, over multiple clusterings obtained by random initializations of the K-means algorithm. This produces a mapping of the clusterings into a new similarity measure between patterns. The final data partition is obtained by applying the single-link method over this similarity matrix. In this paper we further explore and extend this idea, by proposing: (a) the combination of multiple K-means clusterings using variable k; (b) using cluster lifetime as the criterion for extracting the final clusters; and (c) the adaptation of this approach to string patterns