We consider conflict-free colorings of graph neighborhoods: Each vertex of the graph must be assigned a color so that for each vertex v there is at least one color appearing exactly once in the neighborhood of v. The goal is to minimize the number of used colors. We consider both the case of closed neighborhoods, when the neighborhood of a node includes the node itself, and the case of open neighborhoods when a node does not belong to its neighborhood. In this paper, we study complexity aspects of the problem. We show that the problem of conflict-free coloring of closed neighborhoods is NP-complete. Moreover, we give non-approximability results for the conflict-free coloring of open neighborhoods. From a positive point of view, both problems become tractable if parameterized by the vertex cover number or the neighborhood diversity number of the graph. We present simple algorithms which improve on existing results

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In this paper we consider the Supercube, a new interconnection network derived from the hypercube introduced by Sen in 10]. The Supercube has the same diameter and connectivity of a hypercube but can be realized for any number of nodes not only for powers of 2.We study the capabilities of the Supercube to execute parallel programs using graph{embedding techniques. We show that complete binary trees and bidimensional meshes (with a side length power of 2) are spanning subgraphs of the Supercube. Then we prove that the Supercube is Hamiltonian and, when the number of nodes is not a power of 2, it contains all cycles of length greater than 3 as subgraphs.

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This paper studies the problems of One{to{All and All{to{All Communication in optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on di erent wavelengths. In this paper we consider both single{hop and multihop optical networks. In single{hop networks the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high speed communication. In multihop networks a packet may have to be routed through a few intermediate nodes before reaching its nal destination. In both models we give e cient One{to{All and All{to{All Communication algorithms, in terms of time and number of wavelengths. We consider both networks with arbitrary topologies and particular networks of practical interest. Several of our algorithms exhibit optimal performances.