Improved Bounds for Radio k-Chromatic Number of Hypercube Qn

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 avoidedas 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 hypercubeQn, 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 .


Introduction
Radio coloring is derived from the assignment of radio frequencies channels to a set of transmitters.The frequencies assigned depend on the geographical distance between the transmitters: the closer two transmitters are, the greater the potential for interference between their signals.Thus, when the distance between two transmitters is small, the difference in the frequencies assigned must be relatively large, whereas two transmitters at a large distance may be assigned relatively close frequencies.
Radio k-coloring of a graph is a variation of the channel assignment problem.For a simple connected graph G of order n and diameter q, and a positive integer k with 1 k q, a radio k-coloring f of G is an assignment of non-negative integers to the vertices of G such that for every two distinct vertices u and v of G where d u, v is the distance between u and v in G.The span of a radio k-coloring f, span k f , is the maximum integer assigned to a vertex of G.The radio k-chromatic number of G, denoted by rc k G , is defined by rc k G min{span k f : f is a radio k-coloring of G}.Since rc 1 G is the chromatic number χ G , radio k-colorings are a generalization of ordinary vertex coloring of graphs.In the literature, rc q G is termed as radio number of G whereas rc q−1 G and rc q−2 G are called antipodal and nearly antipodal number of G, respectively.Since the problem is to find a radio k-coloring with minimum span, we use the least color 0 in every radio k-coloring appearing in this paper.
Finding radio k-chromatic number of a graph is an interesting yet difficult combinatorial problem with potential applications to FM channel assignment.

Improved Lower Bound
In this section, we give an improved lower bound of rc k Q n for 2 k n−2.For a hypercube Q n of dimension n, the vertex set can be taken as binary n-bit strings and two vertices being adjacent if the corresponding strings differ at exactly one bit.

Definition 2.1. For any two n-bit binary strings a a
between a and b is the number of bits in which they differ.In particular, if x, y ∈ {0, 1}, then d H x, y 0 or 1 according as x y or x / y.
If u and v are two vertices of Q n with a and b as the corresponding strings, then The results in the following lemma may be found in 7 .
Lemma 2.2.For any three vertices x, y, and z of the hypercube Q n , the following holds: The following lower bound for rc k Q n was determined by Kola and Panigrahi 7 .
Theorem 2.3 see 7 .For the hypercube Q n of dimension n 2 and for any positive integer k with 2 k n,

2.1
In the theorem below, we give an improvement of the above lower bound of rc k Q n for all values of n and 2 k n − 2.
Theorem 2.4.For a hypercube Q n of dimension n 2 and for any positive integer k with 2 k n − 2, Proof.Let f be any radio k-coloring of Q n and x 1 , x 2 , . . ., x 2 n be an ordering of the vertices of for every i with 0 i 2 n − 2, we have the following:

2.4
From the above inequality, we have f x i 2 − f x i 3 k 1 − 2n /2 , for all i 1, 3, . . ., 2 n − 3 and summing up these we get, From this inequality and the fact that f x 2 n f x 2 n −1 , we get the result.

Improved Upper Bound
In this section, we give an improved upper bound of rc k Q n .For better presentation, we need the definition below.
Definition 3.1.For two positive integers n and l with l < n, a binary n, l -Gray code is a listing of all the n-bit binary strings such that the Hamming distance between two successive strings is exactly l.Further, a quasi n, l -Gray code is a listing of all the n-bit binary strings such that the Hamming distance between two successive strings is exactly l except between the two items 2 n−1 − 1 and 2 n−1 for which it is l − 1 or l 1.

International Journal of Mathematics and Mathematical Sciences
Notation 3.2.For any positive integer n, we define δ n as

3.1
The following theorem gives a partition of the vertex set of Q n that will be used to find the upper bound of rc k Q n .Theorem 3.3.For a hypercube Q n of dimension n, there exists a partition of the vertex set of Q n with the partite sets U 1 {x i : i 0, 1, . . ., 2 n−1 − 1} and U 2 {y i : i 0, 1, . . ., 2 n−1 − 1} satisfying the followyng properties: where δ n is defined as in Notation 3.2.
Proof.Let Q n−1 and Q n−1 be two copies of Q n−1 induced by all the n-bit strings with starting bit 0 and 1, respectively.Let x 0 , x 1 , . . ., x 2 n−1 −1 be the ordering of the vertices of Q n−1 which induce a quasi n − 1, n/2 -Gray code if n / ≡ 2 mod 4 and a n − 1, n/2 -Gray code if n ≡ 2 mod 4 ; such code exists, see 6 .We take y i x i , i 1, 2, . . ., 2 n−1 − 1, that is, x i is obtained from x i by changing 0s to 1s and vice versa.Now, y 0 , y 1 , . . ., y 2 n−1 −1 is an ordering of the vertices of Q n−1 which induces the same Gray code as in the above.By taking U 1 {x i : i 0, 1, . . ., 2 n−1 − 1} and U 2 {y i : i 0, 1, . . ., 2 n−1 − 1}, we get the result.
The following upper bound for rc k Q n was determined by Kchikech et al. in 12 .
Theorem 3.4 see 12 .For the hypercube Q n of dimension n 2 and for any k 2, The theorem below gives an improvement of this upper bound of rc k Q n .
Theorem 3.5.For the hypercube Q n of dimension n 2 and for any positive integer k, 2 k n−2,

3.3
where δ n is defined as in Notation 3.2.
Proof.Here, we consider the same partition of V Q n as in Theorem 3.3.We first take 2 n/2 − 2 k n − 2. For these values of k, we define a coloring f of V Q n with f x 0 0 and

3.4
We first check the radio condition for the vertex n, radio condition is trivially true for the pair x 2 n−2 , y 2 n−2 .In a similar manner one can show that |f y For the rest of the pairs of vertices in V Q n , checking radio condition is straightforward from the definition of f.However, in Table 1, we compute the values of |f u − f v | and k 1 − d u, v for every pair of vertices u, v Therefore, f is a radio k-coloring of Q n and the span of f is 2 n−1 −1 k 1− n/2 δ n .Next, we consider the value of k as n/2 k 2 n/2 − 3 and define a coloring f with f x 0 0, and

3.5
By a similar way as in the above, one can check that f is a radio k-coloring of Q n with the span f x Observe that the bounds given in Theorems 2.4 and 3.5 agree for k n − 2 with n ≡ 2 mod 4 .Therefore we have determined the nearly antipodal number of Q n which is given in the theorem below.Theorem 3.6.For n ≡ 2 mod 4 , the nearly antipodal number rc

Concluding Remarks
Improved lower and upper bounds of rc k Q n have been obtained in Theorems 2.4 and 3.5, respectively.It is easy to verify that the bound given in Theorem 3.5 is an improvement over the bound in Theorem 3.4.However, below, we check that lower bound given in Theorem 2.4 is really an improvement over the bound in Theorem 2.3.From Theorem 2.4, we have , if k is even.

4.1
For odd integer k,

4.2
Again, if k is an even integer, then

4.3
Theorem 3.6 gives the nearly antipodal number of Q n for n ≡ 2 mod 4 .From Theorems 2.4 and 3.5, one gets that the difference between the lower and upper bound of rc n−2 Q n , n ≡ 0 mod 4 , is equal to one, that is, The concept of radio k-coloring was introduced by Chartrand et al. in 1 .But so far, radio k-chromatic number is known for very limited classes of graphs and for specific values of k only.Radio numbers of C n and P n 2 , C 2 n 3 , and P 2 n 4 are known.Recently, radio numbers of complete m-array trees have been found by Li et al. 5 .Radio number of hypercubes is determined by Khennoufa and Tongi 6 and Kola and Panigrahi 7 independently.Khennoufa et al. have also found antipodal number of Q n .For k n − 1, n − 2, n − 3 and n − 4 the radio k-chromatic number of path P n has been determined in 4, 8-10 , respectively.Juan and liu 11 have determined the antipodal number of cycle C n .Kchikech et al. 12 have given an upper and a lower bound for radio k-chromatic number of cartesian product of two graphs G and G .Also, as a particular case, they have given an upper and a lower bounds for radio k-chromatic number of hypercube Q n , but these bounds were very loose.Recently, Kola and Panigrahi 7 have improved their lower bound for rc k Q n and have shown that this bound is exact for radio number of Q n .In this paper, we obtain an improvement of Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound.Further, we show that these bounds agree for rc n−2 Q n , n ≡ 2 mod 4 .
International Journal of Mathematics and Mathematical Sciences and y i 1 .In this case, we define a coloring f as f x n .Finally, we take 2 k n/2 − 2. From Theorem 2.4, we have d x i , x i 1 , d x i , y i , d x i , y i 1 , d x i 1 , y i , d y i , y i 1 k 1.So, we use the same color for the vertices x i , x i 1 , y i