Cmjv02i06p0882

J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011)

Harmonious Coloring of Honeycomb Networks BHARATI RAJAN, INDRA RAJASINGH and FRANCIS XAVIER D. Department of Mathematics, Loyola College, Chennai, India ABSTRACT A harmonious coloring of a graph is the least number of colors in a vertex coloring such that each pair of colors appears on at most one edge. In this paper we provide an approximation algorithm to obtain the harmonious chromatic number of honeycomb networks.

Keywords: Labeling, harmonious coloring, honeycomb networks.

1. INTRODUCTION Graph labelings, where the vertices are assigned certain values subject to some conditions, have often been motivated by practical problems. Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Graph coloring is a special case of graph labeling; it is an assignment of labels, traditionally called colors, to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Graph coloring enjoys many practical applications as well as theoretical

challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself.

Scheduling, register allocation in compilers, bandwidth allocation, and pattern matching are some of the applications of graph coloring.

2. AN OVERVIEW OF THE PAPER A proper vertex coloring of a graph , is defined as a vertex coloring form a set of k colors such that no two adjacent vertices share a common color.

That is, a k- coloring of the graph , is a mapping : 1,2, ‌ such that , : .

A harmonious coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number h(G) is the least integer k for which G admits a harmonious coloring with k colors.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011)

Harmonious coloring was first proposed by Harary et al.8 in 1982.

However, the proper definition of this notion is due to Hopcroft et al.9 in 1983. In 2004, Campbell et al.5 obtained new lower bounds for h(G) in terms of the independence number. Bodlaender4 provides a proof that establishes the NP-completeness of the harmonious coloring problem for disconnected interval graphs and cographs.

Asdre et al.1 extended Bodlaender’s results by showing that the problem remains NPcomplete for connected interval graphs.

Additionaly, the NP-completeness of the problem has been proved for trees and disconnected bipartite permutation graphs6, 7, connected bipartite permutation graphs and disconnected quasi-threshold graphs2. Since the problem of determining the harmonious chromatic number of a connected cograph is trivial, Asdre et al.2 showed that the harmonious colouring problem is polynomially solvable on connected quasithreshold and threshold graphs.

At this point it is worth noticing that the problem is even harder when restricted to many graph families in which NP-hard problems usually become tractable. There are only few families for which we can have exact solutions in polynomial time (including paths, cycles, unions of paths and cycles, stars, complete graphs, complete bipartite graphs and threshold graphs).

The following two results provide simple bounds for the harmonious chromatic number of any graph.

Result 1[7]: Let G be a simple connected graph. Then the harmonious chromatic number h(G) satisfies ∆ 1, where ∆ is the maximum degree of vertices of G.

This result arises from the observation that, in any harmonious coloring of a graph, any vertex, and all of its neighbours, must all have distinct colors.

Result 2[7]: If a graph G is harmoniously colored with k colors then | | .

Thus we have the following definitions.

Let m be a positive integer. Keith Edwards7 defined Q(m) to be the least positive integer k such that kC2 ≼ m, and q(m) to be the greatest integer k such that kC2 ≤ m. He also defined R(m) to be Q(m)C2 – m and r(m) to be m – q(m)C2. He calculated that Q(m) = (1+(8m + 1)1/2)/2 and q(m) = (1 + (8m + 1)1/2)/2. Also 0 ≤ R(m) ≤ Q(m) – 1 and 0 ≤ r(m) ≤ q(m).

3. HONEYCOMB NETWORKS A high level honeycomb network can be constructed from a low level one. A unit honeycomb network is a hexagon, denoted by HC(1). Honeycomb network of size 2 denoted HC(2), can be obtained by adding six hexagons around the boundary edges of HC(1). Inductively, honeycomb network HC(n) can be obtained from HC(n – 1) by adding a layer of hexagons around the boundary edges of HC(n – 1). Honeycomb network of sizes 1 to 3 are shown in Figure 1.

Alternatively, the size d of HC(n) is determined as the number of hexagons between the center and boundary of HC(n) (inclusive) and the number of vertices and edges of HC(n) are 6 and 9 " 3 respectively.

We use the level numbering scheme proposed by the Sharieh et al.10 for the honeycomb networks. Each node in HC(n) is

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011)

identified by a pair (x, y), where x denotes the line number in which the node exists, and y denotes the location of the node in the line. A node with the address 1,1 is the first node that exists at line number 1. The node 1,2 refers to the second node that exists at line number 1, and so on.

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It is possible to redraw the honeycomb network so that it takes the shape of bricks in a wall network as in Figure 3. This diagram is called the brick drawing of honeycomb.

4. HARMONIOUS COLORING OF HONEYCOMB NETWORKS HC(n) In this section we provide bounds on the harmonious chromatic number of honeycomb networks.

HC(1)

HC(2)

Proposition 1[7]: For any graph G with m edges, $ % , where

HC(3)

$ % &

Figure 1: Honeycomb networks

1,1

Line 1 2,1

Line 2 3,1

Line 3

3,2

2, 2 3,3

1, 3

1,2 2, 3

3,4

2,4 3,5

1,4 2,5

3,6

1,5

2,6 3,7

1,6 2,7

3,8

భ ൗమ '.

Since the number of edges of HC(n) is 9 " 3 , using Proposition we have the following lower bound.

1,7 2,9 2,8 3,9

Theorem 1 (Lower bound): Let G be a 3,10

3,10

Figure 2: Addressing in HC(3)

HC(n). Then

√ మ

The next theorem provides an upper bound for the harmonious chromatic number of HC(n).

Theorem 2 (Upper bound): Let G be a

1,1

Line 1 2,1

Line 2

1,4

1,6

2,3

2,5

2,7

2,4

2,6

3,5

3,3 3,2

1,7

1,2

2,2

Line 3 3,1

1,5

1,3

3,4

2,8

3,7 3,6

2,9

3,9 3,8

3,11 3,10

HC(n). Then We next provide algorithm of HC(n).

మ

.

an

approximation

Algorithm Harmonious: Input: A simple connected graph G with an addressing scheme proposed by Sharieh et al.10.

Figure 3: Addressing nodes in the brick drawing of HC(3)

Step 1: Fix k.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011)

Case (1): 1 .

The first vertex in this line has color

" 1 1.

The alternate

vertices in this line beginning with the first are colored consecutively. The second vertex

in line k has color 2.

Alternate vertices in this line beginning with the second are colored consecutively.

Step 2: Stop when Step 1 cannot be implemented further.

Output: A harmonious coloring of HC(n).

1 5

7 7

11

Case (2): 1 2 .

The first vertex in this line has color 2 "

" 1 " 1. Alternate

vertices in this line beginning with the first are colored consecutively. The second vertex in line k has color

2 " " " 2.

Alternate vertices in this line beginning with the second are colored consecutively. n(k-1)+ (k(k-1)/2)+1

17 9

13 19

16

17 23

6

19 11

10 24

13

15

14

Figure 3: A harmonious coloring of HC(3)

n(k-1)+ (k(k-1)/2)+2

nk+(k(k-1)/2)+k+2

5

18 9

22 12

13

4

8

15 21

12

3

7

14 20

11

10 2

1

9 14

13

12 18

4 8 8

12

11

10

3

2 6 6

nk+(k(k-1)/2)+k nk+(k(k-1)/2)+2k+n

nk+(k(k-1)/2)+k+3

Figure 4: Illustration for case (i) (horizontal edges)

Proof of correctness of Algorithm Harmonious: We now prove that the above coloring is a harmonious coloring.

Case (i): 1 " 1 Consider the line k. The colors are depicted in Figure 5.

The vertices in line k beginning with the first are colored consecutively and vertices beginning with the second are colored consecutively.

Hence the pairs 1 ŕŻžáˆşŕŻžŕŹżŕŹľáˆť ଶ ŕŻžáˆşŕŻžŕŹżŕŹľáˆť ଶ ŕŻžáˆşŕŻžŕŹżŕŹľáˆť ଶ

2 ,

ŕŻžáˆşŕŻžŕŹżŕŹľáˆť

2 ‌

ଶ

1,

2, 1

ଶ ŕŻžáˆşŕŻžŕŹżŕŹľáˆť ଶ

ŕŻžáˆşŕŻžŕŹżŕŹľáˆť

2 ,

are distinct.

Now we consider vertical edges between the lines k and k + 1. The difference in colors on the vertices on the vertical edges is a constant, namely 2n + 2k + 2.

Hence the pairs( " 1

), 2 )*,

are distinct.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

1 )

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011) n(k-1)+ (k(k-1)/2)+1

nk+(k(k+1)/2)+1

n(k-1)+ (k(k-1)/2)+2

nk+(k(k-1)/2)+k

nk+(k(k-1)/2)+k+2

nk+(k(k-1)/2)+2k+n

nk+(k(k+1)/2)+2

nk+(k(k+1)/2)+n+k

nk+(k(k+1)/2) +n+k+3

886

nk+(k(k+1)/2) +n+k+4

nk+(k(k+1)/2)+n+k+1

nk+(k(k+1)/2) +2n+2k+2

Figure 5: Illustration for case (i) (vertical edges)

Case (ii): n(k-1)+ (k(k-1)/2)+2

n(k-1)+ (k(k-1)/2)+1

nk+(k(k-1)/2)+k

nk+(k(k-1)/2)+k+2

nk+(k(k-1)/2)+2k+n

2n(k-n)((k-n)(k-n-1)/2)+n+k+3

2n(k-n)-((k-n)(k-n-1)/2) +2n+2k+1

2n(k-n)((k-n)(k-n-1)/2)+1

2n(k-n)((k-n)(k-n-1)/2)+n+k

2n(k-n)((k-n)(k-n-1)/2)+2

Figure 6: Illustration for case (ii)

Case (iii): 1 2 1 2n(k-n)2n(k-n)-((k-n-1)(k-n-2)/2) ((k-n-1)(k-n-2)/2)+k-n+3 +k-n+2 2n(k-n-1)((k-n-1)(k-n-2)/2)+1

2n(k-n-1)((k-n-1)(k-n-2)/2)+2 2n(k-n)((k-n)(k-n-1)/2)+n+k+3

2n(k-n)((k-n)(k-n-1)/2)+1

2n(k-n)-((k-n-1)(k-n-2)/2) +2n+1 2n(k-n)2n(k-n)((k-n-1)(k-n-2)/2)+n-k ((k-n-1)(k-n-2)/2)+n-k+1 2n(k-n)((k-n)(k-n-1)/2)+4n+1

2n(k-n)((k-n)(k-n-1)/2)+2

2n(k-n)((k-n)(k-n-1)/2)+3n-k

Figure 7: Illustration for case (iii) Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 882-887 (2011)

The proofs for Cases (ii) and (iii) are similar to that of Case (i).

The vertex 2 , 2 has the label This is an upper bound for h(G).

మ

.

5.

In this paper we have provided an upper bound for the harmonious chromatic number of honeycomb networks. The same problem is under investigation to hexagonal networks.

6.

5. CONCLUSION

7.

REFERENCES 1. Asdre K, Ioannidou K and Nikolopoulos S. D, “The harmonious coloring problem is NP- complete for interval and permutation graphs”, Discrete Applied Math, 155, 2377-2382, (2007).

2. Asdre K and Nikolopoulos S.D, “NPcompleteness results for some problems on subclasses of bipartite and chordal graphs”, Theoret. Comput. Sci, 381, 248-259, (2007).

3. Bondy J. A and Murty U.S.R., “Graph theory with Applications”, London: MacMillan (1976).

4. Bodlaender H. L, “Achromatic number is NP- complete for cographs and

8.

9.

10.

interval graphs”, Inform. Proc. Lett, 31, 135 – 138, (1989).

Campbell D and Edwards K. J, “A new lower bound for the harmonious chromatic number”, Australasian Journal of Combinatorics, 29, 99–102, (2004).

Edwards K. J and McDiarmid C, “The complexity of harmonious coloring for trees”, Discrete Applied Math, 57, 133144, (1995).

Edwards K.J, “The harmonious chromatic number and the achromatic number”, Surveys in Combinatorics, 1347, (1997).

Frank O, Harary F and Plantholt M, “The line distinguishing chromatic number of a graph”. Ars Combinatorics, 14, 241–252, (1982).

Hopcroft J and Krishnamoorthy M.S, “On the harmonious colouring of graphs”, SIAM J. Algebra Discrete Math, 4, 306-311, (1983).

Sharieh A, Qatawneh M, Almobaideen W, Sleit A, “Hex-Cell: Modeling, Topological Properties and Routing Algorithm”, European Journal of Scientific Research, ISSN 1450-216X Vol. 22 No.2 , 457-468, (2008).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)