Olivier Togni

Irregularity strength of trees

Abstract The main result of this paper establishes that the irregularity strength of any tree with no vertices of degree two is its number of pendant vertices.

Linear and cyclic radio k-labelings of trees

Motivated by problems in radio channel assignments, we consider radio k-labelings of graphs. For a connected graph G and an integer k ≥ 1, a linear radio k-labeling of G is an assignment f of nonnegative integers to the vertices of G such that |f(x)−f(y)| ≥ k+1−dG(x,y), for any two distinct vertices x and y, where dG(x,y) is the distance between x and y in G. A cyclic k-labeling of G is defined analogously by using the cyclic metric on the labels. In both cases, we are interested in minimizing the span of the labeling. The linear (cyclic, respectively) radio k-labeling number of G is the minimum span of a linear (cyclic, respectively) radio k-labeling of G. In this paper, linear and cyclic radio k-labeling numbers of paths, stars and trees are studied. For the path Pn of order n ≤ k+1, we completely determine the cyclic and linear radio k-labeling numbers. For 1 ≤ k ≤ n−2, a new improved lower bound for the linear radio k-labeling number is presented. Moreover, we give the exact value of the linear radio k-labeling number of stars and we present an upper bound for the linear radio k-labeling number of trees.

Irregularity strength of the toroidal grid

Abstract The irregularity strength of G is the smallest possible value of k for which we can assign positive integers not greater then k to the edges of G , in such a way that the sums at each vertex are distinct numbers. We prove that the irregularity strength of the toroidal grid C m × C n is [( mn + 3)/4].

The irregularity strength of circulant graphs

The irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs.

All-to-all wavelength-routing in all-optical compound networks

Abstract We give theoretical results obtained for wavelength routings in certain all-optical networks. In all-optical networks a single physical optical link can carry several logical signals provided that they are transmitted on different wavelengths. An all-to-all routing in a n -node network is a set of n(n−1) simple paths specified for every ordered pair (x,y) of nodes. The routing will be feasible if an assignment of wavelengths to the paths can be given such that no link will carry in the same direction two different paths of the routing on the same wavelength. With such a routing, it is possible to perform the gossiping in one round. The cost of the routing depends on the number of wavelengths it handles. We give the smallest necessary number of wavelength over all possible routings for networks based on certain compound graphs.

Radio k-labelings for Cartesian products of graphs

Frequency planning consists in allocating frequencies to the transmitters of a cellular network so as to ensure that no pair of transmitters interfere. We study the problem of reducing interference by modeling this by a radio k-labeling problem on graphs: For a graph G and an integer k ≥ 1, a radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that |f(x)−f(y)| ≥ k+1−dG(x,y), for any two vertices x and y, where dG(x,y) is the distance between x and y in G. The radio k-chromatic number is the minimum of max{f(x)−f(y):x,y ∈ V(G)} over all radio k-labelings f of G. In this paper we present the radio k-labeling for the Cartesian product of two graphs, providing upper bounds on the radio k-chromatic number for this product. These results help to determine upper and lower bounds for radio k-chromatic numbers of hypercubes and grids. In particular, we show that the ratio of upper and lower bounds of the radio number and the radio antipodal number of the square grid is asymptotically [3/2].

Autonomic Brokerage Service for an End-to-End Cloud Networking Service Level Agreement

Today, cloud networking which is the ability to connect the user with his cloud services and to interconnect these services within an inter-cloud approach, is one of the recent research areas in the cloud computing research communities. The main drawback of cloud networking consists in the lack of Quality of Service (QoS) assurance and management in conformance with a corresponding Service Level Agreement (SLA). In this paper, we propose a framework for self-establishing an end-to-end service level agreement between a Cloud Service User (CSU) and multiple Cloud Service Providers (CSPs) in a cloud networking environment using brokerage service. We focus on QoS parameters for NaaS and IaaS services. This framework ensures a self-establishing of the proposed SLA using autonomic cloud managers. Moreover, videoconferencing is a very demanding application in terms of bandwidth, delay and CPU resources that we have to guarantee. Therefore, we propose to use our cloud networking architecture and framework to provide the cloud videoconferencing application with QoS guarantee. We simulate the corresponding scenario using the CloudSim toolkit to evaluate its performance. The results reveal a high scalability, and a good video conferencing delay.

S -packing colorings of cubic graphs

Given a non-decreasing sequence

Irregularity strength and compound graphs

S=(s\_1,s\_2, \ldots, s\_k)

Broker and federation based Cloud networking architecture for IaaS and NaaS QoS guarantee

of positive integers, an {\em