Improved Bounds for Radio k

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (nonnegative integers) to the stations in an optimal way such that interference is avoided as reported by Hale (2005). Radio k-coloring of a graph is a special type of channel assignment problem. Kchikech et al. (2005) have given a lower and an upper bound for radio k-chromatic number of hypercube Qn, and an improvement of their lower bound was obtained by Kola and Panigrahi (2010). In this paper, we further improve Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound. Also, our bounds agree for nearly antipodal number of Qn when n≡2 (mod 4)

A Lower Bound for Radio kk-chromatic Number of an Arbitrary Graph

Radio $k$-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph $G$ with diameter $d$ and apositive integer $k$, $1\leq k \leq d$, a radio $k$-coloring of $G$ is an assignment $f$ of positive integers to the vertices of $G$ such that $|f(u)-f(v)|\geq 1+k-d(u, v)$, where $u$ and $v$ are any two distinct vertices of $G$ and $d(u, v)$ is the distance between $u$ and $v$.In this paper we give a lower bound for the radio $k$-chromatic number of an arbitrarygraph in terms of $k$, the total number of vertices $n$ and apositive integer $M$ such that $d(u,v)+d(v,w)+d(u,w)\leq M$ for all $u,v,w\in V(G)$. If $M$ is the triameter we get a better lower bound. We also find the triameter $M$ for several graphs, and show that the lower bound obtained for these graphs is sharp for the case $k=d$

Radio Labelings of Distance Graphs

A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.Comment: 14 page

Radio k-Labelings for Cartesian Products of Graphs

International audienceFrequency 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]

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 $|f(x)-f(y)| ≥ k+1-d_G(x,y)$, 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]