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Linear and cyclic radio k-labelings of trees

100%

Kchikech M., Khennoufa R., Togni O.

Discussiones Mathematicae Graph Theory

2007

tom 27

nr 1

105-123

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 $|f(x)-f(y)| ≥ k+1-d_G(x,y)$, for any two distinct vertices x and y, where $d_G(x,y)$ 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.

Radio k-labelings for Cartesian products of graphs

100%

Kchikech M., Khennoufa R., Togni O.

Discussiones Mathematicae Graph Theory

2008

tom 28

nr 1

165-178

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 $|f(x)-f(y)| ≥ k+1-d_G(x,y)$, for any two vertices x and y, where $d_G(x,y)$ 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].

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Kchikech M.

2 Khennoufa R.

2 Togni O.

1 2008

1 2007

Wyniki wyszukiwania

Linear and cyclic radio k-labelings of trees

Radio k-labelings for Cartesian products of graphs