On 1-Harmonious Chromatic Number of Certain ...joics.org/gallery/ics-1911.pdf · On 1-Harmonious...

On 1-Harmonious Chromatic Number of Certain Interconnection

Networks

Antony Nelson

Department of Mathematics, Loyola College, Chennai 600034, India

[email protected]

Abstract

Graph coloring is one of the significant concepts in graph theory and implemented in many real-time

circumstances. The domain of graph colorings has developed into one of the most desired areas of

graph theory. It is a basic and important problem in scientific reckoning and engineering layout.

As the field of graph coloring is one of the thriving branches of graph theory, many new properties,

speculations, proofs, and algorithms are established, developed and examined by mathematicians

all over the globe. A 1-harmonious coloring is a minimum vertex coloring in which no two incident

edges share the same color pair. In this paper, we investigate the 1-harmonious chromatic number of

triangular tessellation sheet and its induced subgraph, enhanced mesh network, generalized prism,

and generalized fat-tree networks.

AMS Classification Number: 05C15

Keywords : Coloring; 1-harmonious coloring; networks.

1 Introduction

Graph coloring is one of the best known, most significant, popular, and vastly researched sub-

jects in the field of graph theory. It has many applications and hypothesis which were studied

by mathematicians and computer scientists throughout the world. An affirmation of this can be

found in various research articles and books, in which the coloring is studied, and the problems

and conjectures associated with this field of research are being described and solved. The field

of graph coloring, and mathematical problems correlated with this field of the study, enchanted

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mathematicians for a long time [7]. The Vertex Coloring Problem requires a color be assigned to

each vertex such that colors on adjacent vertices are different and the number of colors used to

be minimized. Graph vertex coloring is one of the most studied NP-hard combinatorial optimiza-

tion problems. Research in graph coloring heuristics is very active and improved results have been

obtained recently, notably for coloring very large graphs [8]. It has received much attention in

the literature [2, 6, 11–13, 16], not only for its real-world applications in many engineering fields,

including, among many others, scheduling [14], timetabling [26], register allocation [5], train plat-

forming [3], frequency assignment [9] and communication networks [28], but also for its theoretical

aspects and its difficulty from the computational point of view. Exact algorithms proposed for the

vertex coloring problem can consistently solve only small randomly generated instances, with up to

80 vertices. On the other hand, real-world applications commonly deal with graphs of hundreds or

thousands of vertices, for which the use of heuristic and meta-heuristic techniques is necessary [17].

A harmonious coloring of a graph G [20] is a vertex coloring in which adjacent vertices receiving

different colors and all edges receiving different color pairs and their chromatic number denoted

by h(G). Later Wang et al. [25] generalized harmonious coloring problem as the local harmonious

coloring problem. It restricts the different color pair requirements only needed to satisfied for every

edge within distance d for any vertex and its chromatic number denoted by hd(G) is the least k

such that G has a d-harmonious coloring. Clearly, the original harmonious coloring problem is

dx/2e-harmonious coloring problem if x is the diameter of G. The 1-harmonious coloring [25] is a

kind of vertex coloring such that the color pairs of end vertices of every edge are different only for

adjacent edges and the optimal constraint that the least number of colors be used. Besides, it was

proved [25] that solving the 1-harmonious coloring problem h1(G) is NP-complete and it has been

solved for path and cycle graphs where as the tight bounds for other graphs have been stated as

open problems. Recently, Gao [10] has presented three algorithms to give the coloring procedure

for the 1-harmonious coloring problem but these algorithms were based on an exhaustive search of

vertices and branching rules.

The rest of the paper is organized as follows: In section 2, We have presented a lower bound for

1-harmonious coloring number h1(G) of graph G, where ∆(G) is the maximum degree of the graph

G. We show that this bound is sharp for enhanced mesh, triangle tessellation sheet and its induced

subgraph. In Section 3 and 4, we find h1(G), where G is the generalized prism or the generalized

fat tree network. Finally, we conclude the paper in Section 5.

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2 Mesh Based Networks

In this section, we present a lower bound for 1-harmonious coloring number and prove that the

bound is sharp for some of the mesh based networks such as triangular tessellation sheet and its

induced subgraph, enhanced mesh.

Theorem 2.1. [18] Let G be an any simple connected graph and k be the number of vertices which

have ∆(G) common neighbors. Then h1(G) ≥ ∆(G) + k.

Theorem 2.2. Let G be a simple connected graph and H be any induced subgraph of G, then

h1(G) ≥ h1(H).

Proof. Consider a 1-harmonious coloring of G with h1(G) colors. This induces a 1-harmonious

coloring on H. Hence h1(H) ≤ h1(G).

2.1 Triangular Tessellation Sheet and its Induced Subgraph

A regular tessellation is a pattern made by repeating a regular polygon. In geometrical terminol-

ogy a tessellation is the pattern resulting from the arrangements of regular polygons to cover a

plane without any interstices (gaps) or overlapping. The patterns are usually repeating. Regu-

lar tessellations were made up entirely of congruent regular polygons all meeting vertex to vertex.

There are only three regular tessellations which use a network of equilateral triangles, squares and

hexagons [19]. A triangular grid formed by tiling the plane regularly with equilateral triangles. The

triangular grid Tn, n ≥ 1, is the lattice graph obtained by interpreting the order-(n+ 1) triangular

grid as a graph, with intersection of grid lines being the vertices and the line segments between

vertices being the edges [27], see Figure 1(b). A family of 6-regular graphs, called hexagonal meshes

or H-mesh HX(n) with 2n− 1 levels, that are a multiprocessor interconnection network based on

regular triangular tessellations, see Figure 1(a). Processing nodes on the periphery of an H-mesh

are first wrapped around to make regularity and homogeneity. Hexagonal meshes (or H-meshes)

is a multiprocessor architecture that possesses all the foregoing salient features. A large number of

data manipulation applications need the PN’s on the hexagonal periphery be wrapped around to

meet regularity and homogeneity such that identical software and protocols be applied uniformly

over the network [4].

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l1

l2

l3

l

l5

4

(a)

l1

l2

l3

l4

l5

(b)

Figure 1: Representation of levels in (a) HX(3), (b) T5

Theorem 2.3. The 1-harmonious coloring number of triangular grid network is given by h1(Tn) =

∆(G) + 1.

We next propose a coloring algorithm to find the exact 1-harmonious coloring number of trian-

gular grid network. The edges of a triangle tessellation sheet be partitioned into horizontal edges,

acute edges, and obtuse edges as shown in Figure 2 [23].

Acute edge

Obtuse edge

Horizontal

edge

Figure 2: A triangle tessellation sheet

Procedure 1-HARMOIOUS COLORING (Triangular grid network (Tn))

Input: Triangular grid network.

Algorithm:

(i) Color l1, the level one vertices from left to right with consecutive numbers beginning from

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1 with numbers taken modulo 7.

(ii) Color l2, the level two vertices such that every obtuse edge incident at level l1 will incre-

mented by 2 and every acute edge will be incremented by 3.

(iii) Inductively, proceed level-wise till all vertices are colored.

End 1-HARMOIOUS COLORING

Output: h1(Tn) = 7, see Figure 3(b).

Proof of Correctness: Two vertices adjacent at a particular level ‘l’ have consecutive coloring say i

and i+ 1. The acute edge incident at vertex colored i coincides with obtuse edge incident at vertex

colored i+ 1. By our algorithm, by virtue of an obtuse edge being colored at the other end is i+ 2

and acute edge incident at vertex colored i+ 1 receives the color (i+ 1) + 2 = i+ 3. Therefore the

coloring is well-defined.

Theorem 2.4. Let G be a hexagonal grid. Then h1(G) = ∆(G) + 1.

Proof. Since G is an induced subgraph of triangular tessellation sheet, by the Theorem 2.3, it results

that h1(G) = 7 as shown in Figure 3(a).

2 3 4 5

76 1 2 3

3 4 5 6 7 1

7 1 2 3 4 5 6

5 6 7 1 2 3

3 4 5 6 7

1 2 3 4

(a)

5

2 3

6 7 1

3 4 5

71 2

45 6

1 2 3 4 5 6 7

7 12

34

6

(b)

Figure 3: (a) h1(HX(4)), (b) h1(T7) = 7

Theorem 2.5. Let G be a triangle tessellation sheet. Then G admits 1-harmonious coloring with

the chromatic number h1(G) = ∆(G) + 1.

Proof. Since the proof follows the similar argument of Therom 2.3, we conclude that h1(G) =

∆(G) + 1.

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2.2 Enhanced Mesh Network

An enhanced mesh EN(m,n) [24] is obtained by replacing each 4-cycle of M(m,n) by a wheel, the

hub of the wheel being a new vertex. Let hij , 1 ≤ i ≤ m− 1, 1 ≤ j ≤ n− 1 , be the hub vertices. In

Figure 4, the enhanced mesh EN(5, 5) is illustrated. The edges of the enhanced mesh be partitioned

into horizontal edges, vertical edges, acute edges, and obtuse edges as shown in Figure 4 [23]. In

this section, we prove that the bound is sharp for EN(m,n).

h41 h4 4

h14

h11

Acute edge

Obtuse edge

Horizontal

edge

Vertical

edge

v11

v31

v55

v51

v35

v15

Figure 4: An enhanced mesh EN(5, 5)

Theorem 2.6. For m,n > 2, let G be an enhanced mesh EN(m,n). Then G admits 1-harmonious

coloring with the chromatic number h1(G) = ∆(G) + 1.

Proof. Now, we propose a coloring algorithm to find a 1-harmonious coloring number of EN(m,n).

Procedure 1-HARMOIOUS COLORING (Enhanced mesh)

Input: Enhanced mesh, EN(m,n),m, n > 2.

Algorithm:

(i) Color v1j , row one vertices of EN(m,n) from left to right with consecutive numbers beginning

from 1 with the numbers taken modulo 9.

(ii) Color the vertices of h1j such that every obtuse edge incident at row one vertex will be

incremented by 2 and every acute edge will be incremented by 3.

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(iii) Color v2j , row two vertices such that every obtuse edge of h1j incident at row two vertex

will be incremented by 2 and every acute edge will increment 3.

(iii) Inductively, repeat (ii) and (iii) till all vertices are colored.

Output: h1(EN(m,n)) = 9, see Figure 5.

Proof of Correctness: Two vertices adjacent at a particular row have consecutive coloring say k and

k + 1. The acute edge incident at vertex colored say k coincides with obtuse edge incident at

vertex colored k + 1. By our algorithm, by virtue of an obtuse edge being colored at the other

end is i+2 and acute edge incident at vertex colored k+1 receives the color (k+1)+2 = k+3.

Therefore the coloring is well-defined.

3 4 5

2

7

7

1 32

6

8 9 1

6 7 19

1

54

2

9 1

3

5

4

8

2 3

86

2 3

7

4 5 6

4 5

6 7

Figure 5: h1(EN(5, 5)) = 9

3 Generalized Prism

The generalized prism [1] can be defined as the cartesian product Cm ×Pn of a cycle on m vertices

with a path on n vertices. Let V (Cm × Pn) = {vij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} as shown in Figure

6(a) and clearly, |V (Cm×Pn)| = mn. From the definition, it is easy to see that for any fixed i, the

vertices vij , 1 ≤ j ≤ n, are arranged in a path say P i. It has various applications and in particular

can be important when constructing large communications networks from smaller ones. It provides

important constructions because the connectivity of the newly created graph is larger than that

of the original (connected) graph, regardless of the permutation used [21]. In this section, we first

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state the natural property of 1-harmonious coloring and then we will compute the 1-harmonious

coloring number for generalized prism with the systematic and elegant way of coloring .

v1

v

2

v

v1

v2

v

v1

v2

v

v1

v2

v

1

21

1

2

3

23

3

3

33

4443

(a)

1

2

3

3

4

5

5

1

2

2

3

4

4

5

1

53

4

1

2

(b)

Figure 6: (a) C4 × P3, (b) h1(C5 × P4) = 5

Theorem 3.1. If m ≥ 3 and n ≥ 2, then the generalized prism Cm × Pn admits 1-harmonious

coloring with chromatic number 5 ≤ h1(Cm × Pn) ≤ 6.

Proof. Here, we propose a coloring algorithm in the following cases to find a 1-harmonious coloring

number of Cm × Pn.

Case 1 : m (mod 5) = 0

Procedure 1-HARMOIOUS COLORING (Generalized prism, Cm × Pn)

Input: Generalized prism, Cm × Pn.

Algorithm:

(i) Color P 1 vertices from the inner circle to an outer circle with the consecutive numbers

beginning from 1 with numbers taken modulo 5.

(ii) Color P 2 vertices such that every edge incident at P 1 vertex will be incremented by 2 in

clockwise direction.

(iii) Inductively, proceed path-wise till all vertices are colored.


Output: h1(C5m × Pn) = 5

Proof of Correctness: Three vertices adjacent at a particular path have consecutive coloring say

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k−1, k and k+1. By our algorithm, clearly, for the vertex colored k, the adjacent vertices which

lie in clockwise and anti-clockwise directions will have the color k + 2 and k − 2 respectively.

Therefore the coloring is well-defined.

Case 2 : m (mod 5) 6= 0

In this case, we propose an algorithm by splitting it into the following ways:

1. m ≡ 0 (mod 3)

2. m ≡ 1 (mod 3) or m ≡ 2 (mod 3).

Case 2.1 : m ≡ 0 (mod 3)

We omit the proof of the case since it follows a similar procedure as in Case 1 with numbers

taken modulo 6.

Case 2.2 : m ≡ 1 (mod 3) and m ≡ 2 (mod 3)

(i) Except for the paths Pm−1 and Pm, repeat the procedure of Case 1 for the remaining paths

with numbers taken modulo 6.

(ii) Color the vertices of Pm−1 with consecutive numbers beginning from 6 when m ≡ 1 (mod

3) and beginning from 2 when m ≡ 2 (mod 3).

(iii) Color the vertices of Pm such that every edge incident at vertex in Pm−1 will be decremented

by 2 when m ≡ 1 (mod 3) and incremented by 2 when m ≡ 2 (mod 3).


Output: h1(Cm × Pn) = 6

Proof of Correctness: Clearly, for the paths P i, 2 ≤ i ≤ m−3, the coloring is well-defined, since

it is similar to that of Case 1. For the remaining paths, it is clear that any three vertices adjacent

at a particular path have consecutive colorings say k − 1, k and k + 1. For the vertex colored

k in Pm−2 (P 1), the adjacent vertices which lie in clockwise and anti-clockwise directions will

have the color k+ 3 and k+ 4 (k+ 2 and k+ 3). For the vertex colored k in Pm−1, the adjacent

vertices which lie in clockwise and anti-clockwise directions will have the color k + 4 or k + 2

(when m ≡ 1 (mod 3) or m ≡ 2 (mod 3)) and k + 3. Similarly, for the vertex colored k in Pm,

the adjacent vertices which lie in clockwise and anti-clockwise directions will have the color

k + 3 and k + 2 or k + 4 (when m ≡ 1 (mod 3) or m ≡ 2 (mod 3)), See Figure 7. Therefore

the coloring is well-defined. Further, the maximum of all the colors used is 6 and the proof is

complete.

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1

2

3

5

6

12

3

4

3

4

5

6

5

6

1

2

1

2

3

4

4

5

6

(a)

1

1 2 3 4

3

4

5

6

5

6

1

2

1234

3

4

5

6

6

1

2

3

4

5

6

(b)

1

1 2 3 4

3

4

5

6

5

6

1

2

1

2

3

4

3456

2

3

4

5

4

5

6

5

6

1

2

(c)

Figure 7: h1(C6 × P4) = h1(C7 × P4) = h1(C8 × P4) = 6

4 Generalized Fat Tree Network

The generalized fat tree GFT (h,m,w) of height h consists of mh vertices in the leaf-level and

switching-nodes in the non-leaf levels. Each non-root has w parent nodes and each non-leaf has

m children. The vertex set of GFT (h,m,w) [22] is given by Vh = {(l, i) : 0 ≤ l ≤ h, 0 ≤ i ≤

mh−lwl − 1}, where l is the level of the node and i denotes position of this node in level l as shown

in Figure 8. For any positive integer p, q and r, we denote pq = [p, p, ..., p︸︷︷︸q-times

] and⟨pq⟩r

= [pq, pq, ..., pq︸︷︷︸r-times

].

Fat trees used to interconnect the processors of a general-purpose parallel supercomputer. A fat-

tree routing network is parametrized not only in the number of processors but also for simultaneous

communication it can support. They are a family of general-purpose interconnection strategies

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which effectively utilize any given amount of hardware resource devoted to communication [15].

In this section, we first state the property that the graph G with k vertices which has ∆(G)

common neighbors. Later, we give an exact 1-harmonious coloring number for generalized fat tree

GFT (h,m,w) by fixing the parameters m and w as m = w, where w ≥ 2.

0 0 0 1 0 2 0 3

1 0 1 1 1 2 1 3

2 0 2 1 2 2 2 3

Figure 8: Vertex representation of GFT (2, 2, 2)

Theorem 4.1. [18] Let G be an any connected graph and k be the number of vertices which have

∆(G) common neighbors. Then h1(G) ≥ ∆(G) + k.

Theorem 4.2. For h > 1, the generalized fat tree network GFT (h,m,w) admits 1-harmonious

coloring with chromatic number h1(GFT (h,m,w)) = 3m.

Proof. SinceGFT (1,m,w) is am-partite graph [18] andm number of vertices which have∆(GFT (1,

m,w)) common neighbors. By Theorem 4.1, we have h1(GFT (1,m,w)) = ∆(GFT (1,m,w)) + m.

We now show that for h > 1, GFT (h,m,w) cannot be colored with ∆(GFT (h,m,w)) + 1 colors

with the illustration of GFT (2, 2, 2). Without loss of generality, let the color of vertex (1, 0) in

GFT (2, 2, 2) be 1 and denoted by c(1, 0) = 1. Then its neighboring vertices (0, 0), (0, 1), (2, 0) and

(2, 1) should be colored from the color set {2, 3, 4, 5} in some order. Again, without loss of gener-

ality, we assume c(2, 0) = 2, c(0, 0) = 3, c(2, 1) = 4 and c(0, 1) = 5. In this case, the vertices (1, 1)

and (1, 2) will be colored in the following ways:

1. c(1, 1) = 2 or 4,

2. c(1, 2) = 3 or 5.

We start with the first case c(1, 1) = 2 and c(1, 2) = 3. Since c(1, 1) = 2, we can color the

vertices (2, 2) and (2, 3) using the color set {1, 4} in some order. Let us assume that c(2, 2) = 1

and c(2, 3) = 4. Similarly, since c(1, 2) = 3, we can color the vertices (0, 2) and (0, 3) using the

color set {1, 5} in some order. But whatever might be the order, the vertex (1, 3) is now adjacent

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to two vertices colored with 1. Therefore, h1(GFT (2, 2, 2)) > 5. In the same manner, we can

prove the other three cases that h1(GFT (h, 2, 2)) > 5 as shown in Figure 9. By Theorem 2.2, we

have h1(GFT (h, 2, 2)) ≥ h1(GFT (2, 2, 2)) ≥ 6 and we now color the vertices of GFT (h, 2, 2) with

six(3m = 3(2) = 6) colors as shown in Figure 10.

1

2

3

4

5

1

2

3

4

5

2 3

1

2

3

4

5

2 3

1 4

1 5

1

2

3

4

5

2 5 1

2

3

4

5

4 3 1

2

3

4

5

4 5

1

2

3

4

5

2 5

1 4

4 1

1

2

3

4

5

4 3

1 2

1 5

1

2

3

4

5

4 5

1 2

1 3

Figure 9: Illustration for h1(GFT (h, 2, 2) > 5

We study the graph GFT (h,m,w),m = w,w > 2. We conclude that h1(GFT (h,m,w)) = 3m by

proceeding with the similar argument of GFT (h, 2, 2). First, we color the vertices of GFT (h,m,w)

in level zero as⟨1, 2, ...,m

⟩mh−1 from left to right. Now, we color the vertices in the remaining levels

using 3m-colors set {1, 2, ..., 3m} as the order described in Table 1.

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Table 1: The 1-harmonious coloring order for the levels of GFT (h,m,w), h > 1

S. No. Level iColor order of level i

1 ≤ i ≤ h–1 i = h

1 i (mod 3) = 0

⟨⟨1, 2, ...,m

⟩ml−1 ,

⟨2, 3, ...,m, 1

⟩ml−1 ,⟨

3, 4, ...,m, 1, 2⟩ml−1 , ...,⟨

m, 1, 2, ...,m− 1⟩ml−1

⟩mh−l−1

⟨1, 2, ...,m

⟩ml−1

2 i (mod 3) = 1

⟨⟨m+ 1,m+ 2, ..., 2m

⟩ml−1 ,⟨

m+ 2,m+ 3, ..., 2m,m+ 1⟩ml−1 ,⟨

m+ 3,m+ 4, ..., 2m,m+ 1,m+ 2⟩ml−1 , ...,⟨

2m,m+ 1,m+ 2, ..., 2m− 1⟩ml−1

⟩mh−l−1

⟨m+ 1,m+ 2,

..., 2m⟩ml−1

3 i (mod 3) = 2

⟨⟨2m+ 1, 2m+ 2, ..., 3m

⟩ml−1 ,⟨

2m+ 2, 2m+ 3, ..., 3m, 2m+ 1⟩ml−1 ,⟨

2m+ 3, 2m+ 4, ..., 3m, 2m+ 1, 2m+ 2⟩ml−1 , ...,⟨

3m, 2m+ 1, 2m+ 2, ..., 3m− 1⟩ml−1

⟩mh−l−1

⟨2m+ 1, 2m+ 2,

..., 3m⟩ml−1

Since all the vertices in each level are independent and each vertex (l, i) has its neighborhoods

only in the levels l + 1 and l − 1, the vertices of any three consecutive levels will be colored with

m distinct colors {1, 2, ...,m}, {m + 1,m + 2, ..., 2m} and {2m + 1, 2m + 2, ..., 3m}, see Figure 10.

Thus, we have h1(GFT (h,m,w)) = 3m, h > 1.

1 2 1 2 1 2 1 2

3 4 4 3 3 4 4 3

5 6 5 6 6 5 6 5

1 2 1 2 1 2 1 2

Figure 10: h1(GFT (3, 2, 2) = 6

5 Concluding Remarks

In this paper, we discussed a lower bound for 1-harmonious coloring that the graph G with k

(k ≥ 1) number of vertices have ∆(G) common neighbors needs at least ∆(G) + k colors. We

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apply this bound for triangle tessellation sheet and its induced subgraph, enhanced mesh network

and generalized fat-tree network to show that the bound is tight. Also, we obtain the nice upper

bound for generalized prism network with the systematic and elegant way of coloring. It would be

more interesting to apply this bound on torii, augmented butterfly network, recursive mesh pyramid

network and so on.

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