Perfect 3-Colorings on 4-Regular Graph of Order 9

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Perfect 3-Colorings on 4-Regular Graph of Order 9

Sk Rabiul Islam and Sayantan Maity Department of Mathematics,

Indian Institute of Technology Patna Bihta, Patna 801103, INDIA.

email:sk.pma16, sayantan.pma17@iitp.ac.in.

(Received on: November 8, 2018) ABSTRACT

We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 4-regular graphs of order 9. A perfect n-coloring of a graph is a partition of its vertex set into n parts A 1 , A 2 , ..., A n such that for all p, q ∈ {1,2, ..., n}, each vertex of A p is adjacent to a pq number of vertices of A q . The matrix A = (a pq ) n×n is called quotient matrix or parameter matrix. The concept of a perfect coloring generalizes the concept of completely regular code introduced by P. Delsarte. In particular, we classify all the realizable parameter matrices of perfect 3-colorings on 4-regular graphs of order 9.

AMS classification: 03E02, 05C15, 68R05

Keywords: Perfect colorings, equitable partition, regular graph.

1. INTRODUCTION

We consider only undirected finite simple graphs. Let G be a connected graph then we define x, y ∈ V (G), d(x, y):= dist(x, y) in G (i.e the minimum number of edges in a path joining x and y in G). The diameter of G, diam(G) = max x,y ∈ 𝑉(𝐺) d(x, y) = r (say).

For X ⊆ V (G), the induced sub graph G[X] is a graph with vertex set X and edge set E(G[X]) = {e = (x, y) ∈ E(G) : x, y ∈ X}.

For x ∈ V (G), G i (x) = {y ∈ V (G): d(x, y) = i}, where i ∈ {1,2,...,r} and G −1 (x) = G r+1 (x) = φ. We will write G(x) instead of G 1 (x).

A connected graph G with diameter r is called distance-regular graph if there exist integers x i , y i , z i , where i ∈ {1, 2, ..., r} such that for every x, y ∈ V (G) and

d(x, y) = i, and z i neighbors of x in G i−1 (y) and y i neighbors of x in G i+1 (y) and

Sk Rabiul Islam, et al., Comp. & Math. Sci. Vol.9 (11), 1728-1736 (2018)

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x i = y 0 - y i - z i . The numbers x i , y i , z i , where i ∈ {1, 2, ..., r} are called the intersection number and the array {y 0 , y 1 , ..., y r−1 ; z 1 , ..., z r } is called the intersection array of the distance-regular graph G.

For a graph G and a positive integer n, the mapping T : V (G) → {1, 2, ...., n} is called a perfect n-coloring with matrix A = (a ij ), where i, j ∈ {1, 2, ..., n}, if it is subjective and for all i, j for every vertex of color i, the number of its neighbors of color j is equals to a ij . The matrix A is called the parameter matrix or quotient matrix of a perfect coloring.

In other words perfect n coloring is the equitable partitions of the vertex set into n disjoint parts.

A non empty set C ⊆ V (G) is called a code. Elements of C are called codewords. The distance of x ∈ V (G) from C is d(x, C):= min {d(x, y): y ∈ C} and the covering radius ρ C := maxx∈V (G) d(x, C) of C. A code C gives a natural partition of V (G) and the partition is Π = {G 0 (C), G 1 (C),...,G ρ C (C)}. For x ∈ V (G), δ ⁱ (x, C):=

|G i (x) ∩ C| is called the outer distribution numbers of C, where i ∈ {1, 2, ..., r}. A code C in the distance-regular graph G is called completely regular code if δ i (x, C) only depends on i and d(x, C). Note that a code C is completely regular iff Π is perfect (ρ C

+ 1)-coloring, see ² . So perfect coloring is a generalization of completely regular codes.

The existence of completely regular codes in graphs is a historical problem in mathematics. In 1973, Delsarte ⁴ conjectured the non-existence of nontrivial perfect codes in Johnson graphs. Therefore, some effort has been made on enumerating the parameter matrices of some Johnson graphs, including J(4, 2), J(5, 2), J(6, 2), J(6, 3), J(7, 3), J(8, 3), J(8, 4), and J(v, 3) (v odd) (see ^1,12,13,14 ). Fon-Der-Flass enumerated the parameter matrices (perfect 2-colorings) of n-dimensional hypercube Q n for n < 24. He also obtained some constructions and a necessary condition for the existence of perfect 2-colorings of the n- dimensional cube with a given parameter matrix (see ^7,8,9 ). Aleiyan and Meherbani in ¹⁰ obtained perfect 3- colorings of cubic graphs on 10 vertices. M. Alaeiyan and H. Karami (see ¹¹ ) obtained perfect 2-colorings of generalized Petersen graph. In this paper we discuss about perfect 3-colorings on 4-regular graphs of order 9.

2. PRELIMINARIES

In ⁶ , it is shown that the number of connected 4-regular graphs with 9 vertices is 16. The graphs are given bellow:

4

7 7

G 1 G 2 G 3 G 4

2 1

3 2

4 4

7 5 6

9

1 2 1 2 1

3 3 3

4 4

5 5 5

6 7

7 8 8

8 6 8 9 7 9 9 6

Sk Rabiul Islam, et al., Comp. & Math. Sci. Vol.9 (11), 1728-1736 (2018)

1730

For perfect 3-colorings, n = 3. We called first color white, second color black and third color red. We generally denote a parameter matrix by A = [

𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 ].

We consider all perfect 3-colorings up to renaming the colors, i.e. identify the perfect 3- color with the matrices

[

𝑒 𝑑 𝑓 𝑏 𝑎 𝑐

ℎ 𝑔 𝑖 ] ; [ 𝑖 ℎ 𝑔 𝑓 𝑒 𝑑 𝑐 𝑏 𝑎

] ; [

𝑎 𝑐 𝑏 𝑔 𝑖 ℎ

𝑑 𝑓 𝑒 ] ; [ 𝑒 𝑓 𝑑 ℎ 𝑖 𝑔 𝑏 𝑐 𝑎

] ; [

𝑖 𝑔 ℎ 𝑐 𝑎 𝑏

𝑓 𝑑 𝑒 ]. (I)

Obtained by switching the colors with the original coloring.

The simplest necessary condition for the existences of perfect 3-colorings of 4-regular connected graph with the matrix A is

a + b + c = d + e + f = g + h + i = 4. (II)

Note that the diagonal elements a, e, i of parameter matrix A could not be 4 (as degree of the degree regular graph is 4).

We mean by eigenvalue of a graph is the eigenvalue of the adjacency matrix of this graph.

4

7 7

G 1 G 2 G 3 G 4

2 1

3 2

4 4

5

7 6

9

1 2 1 2 1

3 3 3

4 4

5 5 5

6 7

7 8 8

8 6 8 9 7 9 9 6

G 5 G 6 G 7 G 8

2 1

3 2

4 4

5

9 9

8

1 2 1 2 1

3 3 3

4 4

7 7 7

9 9

5 8 5

7 6 6 8 5 6 6 8

G 9 G 10 G 11 G 12

2 1

3 2

6 6

4 8 4

5

1 2 1 2 1

3 3 3

7 6

8 8 8

5 4

9 4 5

Sk Rabiul Islam, et al., Comp. & Math. Sci. Vol.9 (11), 1728-1736 (2018)

Proposition 1: If T is perfect coloring of a graph G in n colors then any eigenvalue of T is an eigenvalue of G. (see ³ )

Now, without lost of generality, we can assume that |W | ≤ |B| ≤ |R|, where W, B, R represents white, black, red color respectively.

Proposition 2: Let T is perfect 3-coloring of a graph G with the parameter matrix A = [

𝑎 𝑏 𝑐 𝑑 𝑒 𝑓

𝑔 ℎ 𝑖 ]. Then 1. |W |b = |B|d

2. |W |c = |R|g 3. |B| 𝑓 = |R|h.

Note that |W | + |B| + |R| = |V (G)| = 9 and parameter matrix is symmetric with respect to 0 (i.e a ij = 0 ⇔ a ji = 0).

Lemma 1.1: Let G be connected 4-regular graph with 9 vertices and |W | = 1, |B| = 1, |R|

= 7 then G has no perfect 3-coloring.

Proof: From proposition 2 we have b = d, c = 7g, f = 7h. |W | = 1 gives a = 0 and

|B| = 1 gives e = 0. As c = 7g, 0 ≤ c ≤ 4 and 0 ≤ g ≤ 4 gives g = 0 which imply c = 0. So from condition (II) we get b = 4. Similarly d = 4, f = 0, h = 0, i = 4. So the parameter matrix can only be [ 0 4 0

4 0 0 0 0 4

]. Which represent one white vertex adjacent to four black vertices. But there is only one black vertex. So this parameter matrix is not possible.

So G has no perfect 3- coloring.

Lemma 1.2: Let G be connected 4-regular graph with 9 vertices and |W | = 1, |B| = 2, |R|

= 6 then G has no perfect 3-coloring.

Proof: similar as Lemma 1.1.

Lemma 1.3: Let G be connected 4-regular graph with 9 vertices and |W | = 1, |B| = 3, |R|

= 5, then G has no perfect 3-coloring.

Lemma 1.4: Let G be connected 4-regular graph with 9 vertices. If T is a perfect 3- coloring with the matrix A and |W | = 1, |B| = 4, |R| = 4, then A should be one of the followings [ 0 0 4

0 1 3 1 3 0

], [ 0 0 4 0 2 2 1 2 1

] and [ 0 0 4 0 3 1 1 1 2

].

Proof: From the proposition 2 we have b = 4d, c = 4g, f = h. |W | = 1, |B| = 4, |R| = 4 gives a = 0, 0 ≤ e ≤ 3, 0 ≤ i ≤ 3. As b = 4d, 0 ≤ d ≤ 4 gives b = 0 or 4. For b = 0 gives d = 0, c = 4, g = 1. For various values of e we get the following matrices [ 0 0 4

0 1 3 1 3 0 ], [ 0 0 4

0 2 2

1 2 1 ] and [ 0 0 4 0 3 1

1 1 2 ]. Now for b = 4 gives d = 1, c = 0, g = 0. For various values

1732 of e we get the following matrices [ 0 4 0

1 0 3

0 3 1 ], [ 0 4 0 1 1 2

0 2 2 ] and [ 0 4 0 1 2 1

0 1 3 ]. Now by (I) we can obtain the last three matrices by switching the colors of first three matrices. So we ignore last three.

Lemma 1.5: Let G be connected 4-regular graph with 9 vertices and |W | = 2, |B| = 2, |R|

= 5, then G has no perfect 3-coloring.

Lemma 1.6: Let G be connected 4-regular graph with 9 vertices and |W | = 2, |B| =3, |R|

= 4, then G has no perfect 3-coloring.

Lemma 1.7: Let G be connected 4-regular graph with 9 vertices. If 𝑇 is a perfect 3- coloring with the matrix 𝐴 𝑎𝑛𝑑 |𝑊 | = 3, |𝐵| = 3, |𝑅| = 3, 𝑡ℎ𝑒𝑛 𝐴 should be one of the followings

[ 0 1 3 1 2 1

3 1 0 ], [ 0 2 2 2 0 2

2 2 0 ] , [ 0 2 2 2 2 0

2 0 2 ] and [ 2 1 1 1 2 1 1 1 2 ].

So all possible perfect 3-colorings on connected 4-regular graph with 9 vertices are A 1 = [ 0 0 4

0 1 3

1 3 0 ] , A 2 = [ 0 0 4 0 2 2

1 2 1 ], A 3 = [ 0 0 4 0 3 1

1 1 2 ], A 4 = [ 0 1 3 1 2 1

3 1 0 ] , A 5 = [ 0 2 2 2 0 2 2 2 0 ],

A 6 = [ 0 2 2 2 2 0

2 0 2 ] and A 7 = [ 2 1 1 1 2 1 1 1 2 ] .

Now we list all the eigen values of A 1 , A 2 , A 3 , A 4 , A 5 , A 6 and A 7 in the following table:

And all the eigen values of the graphs G 1 , G 2 ,...,G 16 are listed below:

Matrix λ 1 λ 2 λ 3

A 1 -3.30 0.30 4

A 2 -2 1 4

A 3 -1.30 2.30 4

A 4 -3 1 4

A 5 -2 -2 4

A 6 -2 2 4

A 7 1 1 4

Graph λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 λ 9

G 1 -2 -2 -1.56 -1 -1 0 1 2.56 4

G 2 -3 -2 -1 -1 0 0 1 2 4

G 3 -2.75 -1.80 -1.57 -1 -0.45 0.35 1.25 1.96 4 G 4 -2.62 -1.62 -1.62 -1.30 -0.38 0.62 0.62 2.30 4

G 5 -2 -2 -2 -1 -1 1 1 2 4

G 6 -2 -2 -1.53 -1.53 -0.35 -0.35 1.88 1.88 4 G 7 -2.60 -2 -1.53 -1.18 -0.35 0.52 1.26 1.88 4

G 8 -2.84 -2 -1.51 -1 0 0.51 1 1.84 4

G 9 -2.96 -2.25 -1.35 -0.55 0 0.57 0.80 1.75 4 G10 -2.88 -2.26 -1.52 -0.65 0.18 0.53 1 1.60 4

G11 -2.56 -2 -2 -1 0 1 1 1.56 4

G12 -3.30 -1.62 -1.62 -0.62 0.30 0.62 0.62 1.62 4

Sk Rabiul Islam, et al., Comp. & Math. Sci. Vol.9 (11), 1728-1736 (2018)

Now by proposition (1) the possible parameter matrices of the above graphs are listed below:

Theorem: The parameter matrices of the connected 4-regular graph of order 9 are listed below:

G13 -3.56 -2 -1 0 0 0.56 1 1 4

G14 -2 -2 -2 -2 1 1 1 1 4

G15 -3 -2 -2 0 0 1 1 1 4

G16 -2.88 -2.88 -0.65 -0.65 0.53 0.53 1 1 4

Graph A 1 A 2 A 3 A 4 A 5 A 6 A 7

G 1 ×  × ×  × 

G 2 ×  ×    

G 3 × × × × × × ×

G 4 × ×  × × × ×

G 5 ×  × ×   

G 6 × × × ×  × ×

G 7 × × × ×  × ×

G 8 ×  × ×  × 

G 9 × × × × × × ×

G10 × × × × × × 

G11 ×  × ×  × 

G12  × × × × × ×

G13 ×  × ×   

G14 ×  × ×  × 

G15 ×  ×    

G16 × × × × × × 

Graph A 1 A 2 A 3 A 4 A 5 A 6 A 7

G 1 ×  × × × × ×

G 2 × × × × × × ×

G 3 × × × × × × ×

G 4 × ×  × × × ×

G 5 ×  × ×   ×

G 6 × × × ×  × ×

G 7 × × × ×  × ×

G 8 × × × × × × ×

G 9 × × × × × × ×

G10 × × × × × × ×

G11 ×  × × × × ×

G12  × × × × × ×

G13 × × × × × × ×

G14 ×  × ×  × 

G15 ×  ×  × × 

G16 × × × × × × 

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Proof: • We know that A 2 , A 5 and A 7 are the only possible parameter matrix for G 1 . Now we consider mapping T 1,2 as

S 1,2 (5) = 1

S 1,2 (3) = S 1,2 (4) = S 1,2 (6) = S 1,2 (7) = 2 S 1,2 (1) = S 1,2 (2) = S 1,2 (8) = S 1,2 (9) = 3.

It is clear that S 1,2 is the perfect 3-colorings of G 1 with the parameter matrix A 2 . claim: A 5 cannot be parameter matrix. If A 5 is possible then |W | = |B| = |R| = 3. As a = e = i = 0, so we have to choose three disjoint set of three vertices which are non adjacent to each other. The vertex 1 is non adjacent with vertex 6,7,8,9 and 6,7,8,9 forms

K 4 so perfect 3 colorings is not possible for A 5 .

claim: A 7 cannot be parameter matrix. If A 7 is possible then |W | = |B| = |R| = 3. As a = e = i = 2, so we have to choose three disjoint triangle. In G 1 there are no such three disjoint triangles.

• We know that A 3 is the only possible parameter matrices for G 4 . Now we consider mapping S 4,3 as

S 4,3 (9) = 1

S 4,3 (1) = S 4,3 (2) = S 4,3 (3) = S 4,3 (4) = 2 S 4,3 (5) = S 4,3 (6) = S 4,3 (7) = S 4,3 (8) = 3.

It is clear that S 4,3 is the perfect 3-colorings of G 4 with the parameter matrix A 3 .

• We know that A 2 , A 5 , A 6 and A 7 are the only possible parameter matrices for G 5 . Now we consider mapping S 5,2 , S 5,5 and S 5,6 as

S 5,2 (6) = 1

S 5,2 (1) = S 5,2 (4) = S 5,2 (5) = S 5,2 (7) = 2 S 5,2 (2) = S 5,2 (3) = S 5,2 (8) = S 5,2 (9) = 3.

S 5,5 (1) = S 5,5 (6) = S 5,5 (7) = 1 S 5,5 (2) = S 5,5 (5) = S 5,5 (8) = 2 S 5,5 (3) = S 5,5 (4) = S 5,5 (9) = 3.

S 5,6 (4) = S 5,6 (5) = S 5,6 (6) = 1 S 5,6 (1) = S 5,6 (2) = S 5,6 (3) = 2 S 5,6 (7) = S 5,6 (8) = S 5,6 (9) = 3.

It is clear that S 5,2 , S 5,5 and S 5,6 are the perfect 3-colorings of G 5 with the parameter matrices A 2 , A 5 and A 6 .

• We know that A 5 is the only possible parameter matrices for G 6 . Now we consider mapping S 6,5 as

S 6,5 (1) = S 6,5 (6) = S 6,5 (7) = 1 S 6,5 (2) = S 6,5 (5) = S 6,5 (8) = 2 S 6,5 (3) = S 6,5 (4) = S 6,5 (9) = 3.

It is clear that S 6,5 is the perfect 3-colorings of G 6 with the parameter matrix A 5 .

• We know that A 5 is the only possible parameter matrix for G 7 . Now we consider

mapping S 7,5 as

Sk Rabiul Islam, et al., Comp. & Math. Sci. Vol.9 (11), 1728-1736 (2018) S 7,5 (1) = S 7,5 (6) = S 7,5 (8) = 1

S 7,5 (2) = S 7,5 (5) = S 7,5 (7) = 2 S 7,5 (3) = S 7,5 (4) = S 7,5 (9) = 3.

It is clear that S 7,5 is the perfect 3-colorings of G 7 with the parameter matrix A 5 .

• We know that A 2 , A 5 and A 7 are the only possible parameter matrices for G 11 . Now we consider mapping S 11,2 as

S 11,2 (7) = 1

S 11,2 (1) = S 11,2 (2) = S 11,2 (5) = S 11,2 (6) = 2 S 11,2 (3) = S 11,2 (4) = S 11,2 (8) = S 11,2 (9) = 3.

It is clear that S 11,2 is the perfect 3-colorings of G 11 with the parameter matrix A 2 .

• We know that A 1 is the only possible parameter matrix for G 12 . Now we consider mapping S 12,1 as

S 12,1 (7) = 1

S 12,1 (1) = S 12,1 (2) = S 12,1 (8) = S 12,1 (9) = 2 S 12,1 (3) = S 12,1 (4) = S 12,1 (5) = S 12,1 (6) = 3.

It is clear that S 12,1 is the perfect 3-colorings of G 12 with the parameter matrix A 1 .

• We know that A 2 , A 5 and A 7 are the only possible parameter matrices for G 14 . Now we consider mapping S 14,2 , S 14,5 and S 14,7 as

S 14,2 (3) = 1

S 14,2 (4) = S 14,2 (5) = S 14,2 (6) = S 5,2 (7) = 2 S 14,2 (1) = S 14,2 (2) = S 14,2 (8) = S 5,2 (9) = 3.

S 14,5 (1) = S 14,5 (7) = S 14,5 (8) = 1 S 14,5 (2) = S 14,5 (4) = S 14,5 (9) = 2 S 14,5 (3) = S 14,5 (5) = S 14,5 (6) = 3.

S 14,7 (1) = S 14,7 (2) = S 14,7 (3) = 1 S 14,7 (4) = S 14,7 (6) = S 14,7 (8) = 2 S 14,7 (5) = S 14,7 (7) = S 14,7 (9) = 3.

It is clear that S 14,2 , S 14,5 and S 14,7 are the perfect 3-colorings of G 14 with the parameter matrices A 2 , A 5 and A 7 .

• We know that A 2 , A 4 , A 5 , A 6 and A 7 are the only possible parameter matrices for G 15 . Now we consider mapping S 15,2 , S 15,4 and S 15,7 as

S 15,2 (3) = 1

S 15,2 (4) = S 15,2 (5) = S 15,2 (6) = S 5,2 (7) = 2 S 15,2 (1) = S 15,2 (2) = S 15,2 (8) = S 5,2 (9) = 3.

S 15,4 (4) = S 15,4 (7) = S 15,4 (9) = 1 S 15,4 (1) = S 15,4 (2) = S 15,4 (3) = 2 S 15,4 (5) = S 15,4 (6) = S 15,4 (8) = 3.

S 15,7 (1) = S 15,7 (4) = S 15,7 (5) = 1

S 15,7 (2) = S 15,7 (6) = S 15,7 (7) = 2

S 15,7 (3) = S 15,7 (8) = S 15,7 (9) = 3.

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It is clear that S 15,2 , S 15,4 and S 15,7 are the perfect 3-colorings of G 15 with the parameter matrices A 2 , A 4 and A 7 .

• We know that A 7 is the only possible parameter matrix for G 16 . Now we consider mapping S 16,7 as

S 16,7 (1) = S 16,7 (2) = S 16,7 (3) = 1 S 16,7 (4) = S 16,7 (7) = S 16,7 (8) = 2 S 16,7 (5) = S 16,7 (6) = S 16,7 (9) = 3.

It is clear that S 16,7 is the perfect 3-colorings of G 16 with the parameter matrix A 7 . REFERENCES

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