Improved Bounds for Radio Chromatic Number of Hypercube

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Volume 2011, Article ID 961649,7pages doi:10.1155/2011/961649

Research Article

Improved Bounds for Radio

k

-Chromatic Number

of Hypercube

Q

n

Laxman Saha, Pratima Panigrahi, and Pawan Kumar

Department of Mathematics, IIT Kharagpur, Kharagpur 721302, India

Correspondence should be addressed to Laxman Saha,[email protected]

Received 13 December 2010; Accepted 18 February 2011

Academic Editor: Brigitte Forster-Heinlein

Copyrightq2011 Laxman Saha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Hale2005. Radio k-coloring of a graph is a special type of channel assignment problem. Kchikech et al.2005have 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 Panigrahi2010. 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 ofQnwhen

n≡2mod 4.

1. Introduction

Radio coloring is derived from the assignment of radio frequencies channelsto 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.

Radiok-coloring of a graph is a variation of the channel assignment problem. For a simple connected graphGof ordernand diameterq, and a positive integerkwith 1kq, a radiok-coloringf ofGis an assignment of non-negative integers to the vertices ofGsuch that for every two distinct verticesuandvofG

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wheredu, vis the distance betweenuandvinG. The span of a radiok-coloringf, spankf,

is the maximum integer assigned to a vertex ofG. The radiok-chromatic number ofG, denoted by rckG, is defined by rckG min{spankf:f is a radiok-coloring ofG}. Since rc1G

is the chromatic number χG, radio k-colorings are a generalization of ordinary vertex coloring of graphs. In the literature, rcqGis termed as radio number ofGwhereas rcq−1G

and rc_q−2G are called antipodal and nearly antipodal number of G, respectively. Since the problem is to find a radiok-coloring with minimum span, we use the least color 0 in every radiok-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. The concept of radiok-coloring was introduced by Chartrand et al. in 1 . But so far, radiok-chromatic number is known for very limited classes of graphs and for specific values ofk only. Radio numbers ofCnandPn2 ,Cn23 , andPn24 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 Tongi6 and Kola and Panigrahi 7 independently. Khennoufa et al. have also found antipodal number ofQn. For k n−1, n−2, n−3 and n−4 the radio

k-chromatic number of pathPnhas been determined in4,8–10 , respectively. Juan and liu

11 have determined the antipodal number of cycleCn. Kchikech et al.12 have given an

upper and a lower bound for radiok-chromatic number of cartesian product of two graphs

GandG. Also, as a particular case, they have given an upper and a lower bounds for radio

k-chromatic number of hypercubeQn, but these bounds were very loose. Recently, Kola and

Panigrahi7 have improved their lower bound for rckQnand have shown that this bound

is exact for radio number ofQn. 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 rcn−2Qn, n≡2 mod 4.

2. Improved Lower Bound

In this section, we give an improved lower bound of rckQnfor 2kn−2. For a hypercube

Qnof dimensionn, the vertex set can be taken as binaryn-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 a0a1· · ·an−1 and b b0b1· · ·bn−1, the

Hamming distancedHa, bbetweenaandbis the number of bits in which they diﬀer. In

particular, ifx, y∈ {0,1}, thendHx, y 0 or 1 according asxyorx /y.

If u and v are two vertices of Qn with a and b as the corresponding strings, then

dQnu, v dHa, b. Twon-bit binary strings may diﬀer in at mostnpositions, so diameter ofQnisn.

The results in the following lemma may be found in7 .

Lemma 2.2. For any three verticesx, y, andzof the hypercubeQn, the following holds:

idx, y dy, z dx, z2n,

iidx, y dy, z dx, z 2n, ifdx, y n.

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Theorem 2.3see7 . For the hypercubeQnof dimensionn2 and for any positive integerkwith

2kn,

rckQn ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

3k1−2n

2

2n−1−k1

2 , if kis odd,

3k1−2n1 2

2n−1−k2

2 , if kis even.

2.1

In the theorem below, we give an improvement of the above lower bound of rckQn

for all values ofnand 2kn−2.

Theorem 2.4. For a hypercubeQnof dimensionn2 and for any positive integerkwith 2k

n−2,

rckQn 3k1−2n

2

2n−1−1. 2.2

Proof. Letfbe any radiok-coloring ofQnandx1, x2, . . . , x2n be an ordering of the vertices of

Qnsuch thatfxj1fxj, 1j2n−1. Obviously,fx1 0 and spanf fx2n. Since

fis a radiok-coloring ofQn, for everyiwith 0i2n−2, we have the following:

fxi1−fxik1−dxi1, xi,

fxi2−fxi1k1−dxi2, xi1,

fxi2−fxik1−dxi2, xi.

2.3

Adding2.3we get,

2fxi2−fxi

3k1−dxi, xi1 dxi2, xi1 dxi2, xi

3k1−2n fromiof Lemma 2.2. 2.4

From the above inequality, we have fxi2 − fxi 3k 1 − 2n/2, for alli

1,3, . . . ,2n−3 and summing up these we get,fx2n₋₁3k1−2n/22n−1−1. From this inequality and the fact thatfx2nfx₂n₋₁, we get the result.

3. Improved Upper Bound

In this section, we give an improved upper bound of rckQn. For better presentation, we

need the definition below.

Definition 3.1. For two positive integersnandlwithl < n, a binaryn, l-Gray code is a listing

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Notation 3.2. For any positive integern, we defineδnas

δn

⎧ ⎨ ⎩

0, if n≡2mod 4,

1, if n /≡2mod 4.

3.1

The following theorem gives a partition of the vertex set ofQnthat will be used to find

the upper bound of rckQn.

Theorem 3.3. For a hypercubeQnof dimensionn, there exists a partition of the vertex set ofQnwith

the partite setsU1{xi:i0,1, . . . ,2n−1−1}andU2 {yi :i0,1, . . . ,2n−1−1}satisfying the

followyng properties:

adxi, xi1 n/2dyi, yi1, for alli0,1, . . . ,2n−1−2 excepti2n−2−1,

bdxi, xi1 n/2 −δnorn/2δn, fori2n−2−1,

cdyi, yi1 n/2 −δnorn/2δn, fori2n−2−1,

ddxi, yi n, for alli0,1, . . . ,2n−1−1,

whereδnis defined as inNotation 3.2.

Proof. LetQ_n−1 andQ_n−1 be two copies ofQn−1 induced by all then-bit strings with starting

bit 0 and 1, respectively. Letx0, x1, . . . , x2n−1₋₁ be the ordering of the vertices ofQ_n−1 which

induce a quasin−1,n/2-Gray code ifn /≡2 mod 4and an−1, n/2-Gray code ifn ≡

2mod 4;such code exists, see 6 . We take yi xi, i 1,2, . . . ,2n−1 −1, that is, xi is

obtained fromxiby changing 0sto 1sand vice versa. Now,y0, y1, . . . , y2n−1₋₁is an ordering of

the vertices ofQ_n−1 which induces the same Gray code as in the above. By takingU1 {xi :

i0,1, . . . ,2n−1−1}andU2{yi:i0,1, . . . ,2n−1−1}, we get the result.

The following upper bound for rckQnwas determined by Kchikech et al. in12 .

Theorem 3.4see12 . For the hypercubeQnof dimensionn2 and for anyk2,

rckQn2n−1k−2n−11. 3.2

The theorem below gives an improvement of this upper bound of rckQn.

Theorem 3.5. For the hypercubeQnof dimensionn2 and for any positive integerk, 2kn−2,

rckQn

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2n−1−1k1−n 2

δn, if 2n

2

−2kn−2,

k2n−2− k 2

δn, ifn

2

−1k2n

2

−3,

k2n−2−k, if 2kn

2

−2,

3.3

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Table 1

u, v∈U1∪U2 Value of|i−j| du, v k1−du, v |fu−fv|

xi, xj∈U1, oryi, yj∈U2 |i−j|1 n/2 k1− n/2 k1− n/2

|i−j|2 1 k k

xi∈U1, yj∈U2

i−j0 n 0 0

|i−j|1 n/2 k1− n/2 k1− n/2

|i−j|2 1 k k

Proof. Here, we consider the same partition ofVQnas inTheorem 3.3. We first take 2n/2−

2kn−2. For these values ofk, we define a coloringfofVQnwithfx0 0 and

fxi fxi−1 k1−

_n

2

, i1,2, . . . ,2n−1−1, excepti2n−2,

fx2n−2 fx₂n−2₋₁ k1−

_n

2

δn,

fyi

fxi, i0,1,2, . . . ,2n−1−1.

3.4

We first check the radio condition for the vertex x2n−2 with all other vertices. Let

v ∈ VQn − {x2n−2}. From the definition of f, |fx₂n−2 − fu| k, except

v ∈ {x2n−2₋₁, y₂n−2₋₁, y₂n−2,x₂n−2₁, y₂n−21}. From Theorem 3.3b and Lemma 2.2ii, we have

dx2n−2₋₁, x₂n−2,dy₂n−2₋₁, x₂n−2n/2 −δnand thereforefx₂n−2−fv k1δn− n/2 k1−dx2n−2, v, forv x₂n−2₋₁ ory₂n−2₋₁. Similarly, forv x₂n−2₁ ory₂n−2₋₁, we

showfx2n−2−fu k1−dx₂n−2, u. Sincefx₂n−2 fy₂n−2and dx₂n−2, y₂n−2 n,

radio condition is trivially true for the pairx2n−2, y₂n−2. In a similar manner one can show

that|fy2n−2−fv|k1−dy₂n−2, v, for allv∈VQ_n− {y₂n−2}. For the rest of the pairs

of vertices inVQn, checking radio condition is straightforward from the definition of f.

However, in Table1, we compute the values of|fu−fv|andk1−du, vfor every pair of verticesu, v∈VQn− {x2n−2, y₂n−2}.

Therefore,fis a radiok-coloring ofQnand the span offis2n−1−1k1−n/2δn.

Next, we consider the value ofkasn/2 k 2n/2 −3 and define a coloringf with

fx0 0, and

fxi1 fxi

k

2

, ifiis even and_{i /}2n−2−1,

fxi1 fxi k

2

, ifiis odd,

fxi1 fxi

k

2

δn, i2n−2−1,

fyi

fxi, i0,1,2, . . . ,2n−1−1.

3.5

By a similar way as in the above, one can check thatf is a radiok-coloring ofQn with the

spanfx2n−1₋₁ 2n−2k/2 2n−2−1k/2δn k2n−2− k/2δn.

Finally, we take 2 k n/2 −2. FromTheorem 2.4, we havedxi, xi1,dxi, yi,

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and yi1. In this case, we define a coloring f as fxi i/2k fyi,fxi1 fxi,

fyi1 fyi,i0,2, . . . ,2n−1−2. It is easy to show thatfis a radiok-coloring ofQnwith

the spank2n−2−k.

Observe that the bounds given in Theorems2.4and3.5agree forkn−2 withn≡2 mod 4. Therefore we have determined the nearly antipodal number ofQnwhich is given in

the theorem below.

Theorem 3.6. Forn ≡ 2mod 4, the nearly antipodal number rcn−2Qnof the hypercubeQn is

2n−1−1n−2/2.

4. Concluding Remarks

Improved lower and upper bounds of rckQnhave been obtained in Theorems2.4and3.5,

respectively. It is easy to verify that the bound given inTheorem 3.5is an improvement over the bound inTheorem 3.4. However, below, we check that lower bound given inTheorem 2.4

is really an improvement over the bound inTheorem 2.3. FromTheorem 2.4, we have

rckQn ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

3k1−2n

2

2n−1−1, ifk is odd,

3k1−2n1 2

2n−1−1, ifk is even,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

3k1−2n

2

2n−1n−3k1

2 , ifk is odd,

3k1−2n1 2

2n−1n−3k4

2 , ifk is even.

4.1

For odd integerk,

3k1−2n

2

2n−1n−3k1

2

−

3k1−2n

2

2n−1−k1 2

n−k−11, if 2kn−2.

4.2

Again, ifkis an even integer, then

3k1−2n1 2

2n−1n−3k4

2

−

3k1−2n1 2

2n−1−k2 2

n−k−11, if 2kn−2.

4.3

Theorem 3.6gives the nearly antipodal number ofQnforn≡2 mod 4. From Theorems2.4

and3.5, one gets that the diﬀerence between the lower and upper bound of rcn−2Qn,n≡0

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