Study on Completely Regular Code and Perfect Colorings on 7-Regular Graph of Order 10

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Study on Completely Regular Code and Perfect Colorings on 7-Regular Graph of Order 10

Sayantan Maity and Sk Rabiul Islam Department of Mathematics,

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

email: sayantanmaity100, [email protected]

(Received on: November 24, 2018)

ABSTRACT

We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 7-regular graphs of order 10. 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 7-regular graphs of order 10.

AMS classification: 03E02; 05C15; 68R05.

Keywords: Perfect colorings, equitable partition, regular graph.

1. DEFINITION AND 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 smallest number of edges in a path joining x and y in G). The diameter of G, diam(G) = maxx,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 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 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

surjective 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 colorings 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 ⊓ = {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 ⊓ 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 ¹⁰ obtained perfect 3- colorings of cubic graphs on 10 vertices. M. Alaeiyan and H. Karami ¹¹ obtained perfect 2-colorings of generalised Petersen graph. In this paper we discuss about perfect 3-colorings on 7-regular graphs of order 10.

2. PRELIMINARIES

In ⁶ , it is shown that the number of connected 7-regular graphs with 10 vertices are 5.

The graphs are given bellow:

G 1 G 2 G 3

G 4 G 5

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 ] .

2 1 2 1 2 1

4 3 4 3 4 3

5 6 7 5 7 5

8 7

9 10

6 8

9 10

6 8

9 10

2 1 2 1

4 3 4 3

7 5 6 5

9 8

10 6

8 7

10 9

G 4 G 5

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 7-regular connected graph with the matrix A is

a + b + c = d + e + f = g + h + i = 7. (II)

Note that the diagonal elements a, e, i of parameter matrix A could not be 7 (as degree of the degree regular graph is 7).

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)| = 10 and parameter matrix is symmetric with respect to 0 (i.e if a ij = 0 ⇔ a ji = 0).

Lemma 1.1: Let G be connected 7-regular graph with 10 vertices. And |W | = 1, |B| = 1,

|R| = 8 then G has no perfect 3-coloring.

Proof: From proposition 2 we have b = d, c = 8g, f = 8h. |W | = 1 gives a = 0 and

|B| = 1 gives e = 0. As c = 8g , 0 ≤ c ≤ 7 and 0 ≤ g ≤ 7 gives g = 0 which imply c = 0.

So from condition (II) we get b = 7. Similarly d = 7, f = 0, h = 0, i = 7. So the parameter matrix can only be [ 0 7 0

7 0 0 0 0 7

] . Which represent one white vertex adjacent to seven black vertices. But there are 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 7-regular graph with 10 vertices. If T is a perfect 3- coloring with the matrix A and |W | = 1, |B| = 2, |R| = 7, then A should be [ 0 0 7

0 0 7 1 2 4 ] . Proof: similar as Lemma 1.1.

Lemma 1.3: Let G be connected 7-regular graph with 10 vertices. And |W | = 1, |B| = 3,

|R| = 6 then G has no perfect 3-coloring.

Lemma 1.4: Let G be connected 7-regular graph with 10 vertices. And |W | = 1, |B| = 4,

|R| = 5 then G has no perfect 3-coloring.

Lemma 1.5: Let G be connected 7-regular graph with 10 vertices. If T is a perfect 3-

coloring with the matrix A and |W | = 2, |B| = 2, |R| = 6, then A should be one of the

followings

[ 0 1 6 1 0 6

2 2 3 ] and [ 1 0 6 0 1 6 2 2 3 ] .

Lemma 1.6: Let G be connected 7-regular graph with 10 vertices. And |W | = 2, |B| = 3,

|R| = 5 then G has no perfect 3-coloring.

Lemma 1.7: Let G be connected 7-regular graph with 10 vertices. If T is a perfect 3- coloring with the matrix A and |W | = 2, |B| = 4, |R| = 4, then A should be one of the followings [ 1 2 4

1 2 4 2 4 1

] , [ 1 2 4 1 3 3

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

] and [ 1 4 2 2 2 3 1 3 3 ] . Proof: similar as above.

Note that, we can obtain the third matrix by switching the colors of the first matrix and fourth matrix by switching the colors of the second matrix. So we ignore third and fourth matrices.

Lemma 1.8: Let G be connected 7-regular graph with 10 vertices. If T is a perfect 3- coloring with the matrix A and |W | = 3, |B| = 3, |R| = 4, then A should be one of the followings [ 0 3 4

3 0 4

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

3 3 1 ] and [ 2 1 4 1 2 4 3 3 1 ] .

Note that in the second matrix a = 1 and |W | = 3, it cannot be possible. So we ignore the second matrix.

So all possible perfect 3-colorings on connected 7-regular graph with 10 vertices are A 1 = [ 0 0 7

0 0 7 1 2 4

] , A 2 = [ 0 1 6 1 0 6 2 2 3

] , A 3 = [ 1 0 6 0 1 6 2 2 3

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

] , A 5 = [ 1 2 4 1 3 3 2 3 2

] ,

A 6 = [ 0 3 4 3 0 4

3 3 1 ] , and A 7 = [ 2 1 4 1 2 4 3 3 1 ] .

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:

Matrix λ 1 λ 2 λ 3

A 1 -3 0 7

A 2 -3 -1 7

A 3 -3 1 7

A 4 -3 0 7

A 5 -1.62 0.62 7

A 6 -3 -3 7

A 7 -3 1 7

And all the eigen values of the graphs G 1 , G 2 , G 3 , G 4 and G 5 are listed below:

Graph λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 λ 9 λ10

G 1 -3 -2 -2 -1 -1 0 0 1 1 7

G 2 -3 -3 -1 -1 0 0 0 0 1 7

G 3 -2.62 -2.62 -1.62 -1.62 -0.38 -0.38 0.62 0.62 1 7

G 4 -3 -2.25 -2.25 -0.55 -0.55 0.80 0.80 0 0 7

G 5 -3 -1.62 -1.62 -1.62 -1.62 0.62 0.62 0.62 0.62 7

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

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 × × × ×   ×

Theorem: The parameter matrices of the connected 7-regular graph of order 10 are listed below:

Proof: We know that A 1 , A 2 , A 3 , A 4 , A 6 and A 7 are the only possible parameter matrices for G 1 . Now we consider mapping T 1,4 and T 1,7 as

T 1,4 (3) = T 1,4 (6) = 1

T 1,4 (4) = T 1,4 (5) = T 1,4 (7) = T 1,4 (8) = 2 T 1,4 (1) = T 1,4 (2) = T 1,4 (9) = T 1,4 (10) = 3.

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

T 1,7 (1) = T 1,7 (2) = T 1,7 (9) = T 1,7 (10) = 3.

It is clear that T 1,4 and T 1,7 are the perfect 3-colorings of G 1 with the parameter matrices A 4 and A 7 respectively.

claim: A 1 is not possible. As |W | = 1, |B| = 2 and b = 0 therefore (W, B, B) = {(1, 9, 10), (2, 9, 10), (3, 7, 8), (4, 6, 8), (5, 6, 7), (6, 4, 5), (7, 3, 5), (8, 3, 4), (9, 1, 2), (10, 1, 2)}. As e = 0 none of these are possible.

similarly we can prove for A 2 , A 3 and A 6 are cannot be parameter matrices.

We know that A 1 , A 2 , A 3 , A 4 , A 6 and A 7 are the only possible parameter matrices for G 2 . Now we consider mapping T 2,1 , T 2,2 , T 2,3 , T 2,4 , and T 2,6 as

T 2,1 (3) = 1

T 2,1 (7) = T 2,1 (8) = 2

T 2,1 (1) = T 2,1 (2) = T 2,1 (4) = T 2,1 (5) = T 2,1 (6) = T 2,1 (9) = T 2,1 (10) = 3.

T 2,2 (1) = T 2,2 (9) = 1 T 2,2 (2) = T 2,2 (10) = 2

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

T 2,3 (1) = T 2,3 (2) = 1 T 2,3 (9) = T 2,3 (10) = 2

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

T 2,4 (3) = T 2,4 (5) = 1

T 2,4 (4) = T 2,4 (6) = T 2,4 (7) = T 2,4 (8) = 2 T 2,4 (1) = T 2,4 (2) = T 2,4 (9) = T 2,4 (10) = 3.

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

T 2,6 (1) = T 2,6 (2) = T 2,6 (9) = T 2,6 (10) = 3.

It is clear that T 2,1 , T 2,2 , T 2,3 , T 2,4 , and T 2,6 are the perfect 3-colorings of G 2 with the pa- rameter matrices A 1 , A 2 , A 3 , A 4 and A 6 respectively. We can prove A 7 cannot be parameter matrix similar as above.

We know that A 5 is the only possible parameter matrix for G 3 . Now we consider mapping T 3,5 as T 3,5 (5) = T 3,5 (10) = 1

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

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 × × × ×  × ×

It is clear that T 3,5 is the perfect 3-colorings of G 3 with the parameter matrix A 5 . We know that A 1 , A 4 and A 6 are the only possible parameter matrices for G 4 . Now we consider mapping T 4,1 as

T 4,1 (4) = 1

T 4,1 (5) = T 4,1 (6) = 2

T 4,1 (1) = T 4,1 (2) = T 4,1 (3) = T 4,1 (7) = T 4,1 (8) = T 4,1 (9) = T 4,1 (10) = 3.

It is clear that T 4,1 is the perfect 3-colorings of G 4 with the parameter matrices A 1 . We can prove A 4 and A 6 cannot be parameter matrices similar as above.

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

T 5,5 (3) = T 5,5 (10) = 1

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

It is clear that T 5,5 is the perfect 3-colorings of G 5 with the parameter matrix A 5 . We can prove A 7 cannot be parameter matrix similar as above.

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