Perfect 3- Coloring of the Cuboctahedral and Chvatal of 4 Regular Graph

Perfect 3- coloring of the cuboctahedral and chvatal of 4 regular graph

Mehdi Alaeiyan and Mozhgan Keyhani

Department of Mathematics, Iran University of Scienece and Technology, 16846, Narmak, Tehran, Iran

February 23, 2021

Abstract

A coloring of the vertex set V of graph G = (V,E) with m colors, is called perfect if all color are used, and for all i, j the number of neighbors colors j of any vertex

V of color i is a constant aij. The matrix A = (aij)i,j∈1,...,m is called the parameter matrix .In this paper we study the Perfect 3- coloring of the cuboctahedral and chvatal 4 regular graph. A cuboctahedral graph is the graph of vertices and edges of the cuboctahedron. It can also be constructed as the line graph of the cube. The chvatal graph is an undirected graph with 12 vertices and 24 edges. First, by applying the condition and limitation of the initial theorems, we identify the possible parameter matrices and then with the help of coloring our graphs, we examine their existence or non- existence. Keywords: Connected graph, Parameter matrices, Perfect coloring, Equitable par- tition.

1 Introduction

Definition 1.1. A vertex coloring of a graph is called perfect if each vertex of color i has exactly aij neighbors of color j.

1 Definition 1.2. For each graph G and each integer m, a mapping T : V (G) → 1, ..., m is called a perfect m- coloring with matrix A = (aij)i,j∈1,...,m if it is surjective, and for all i, j for every vertex of color i, the number of its neighbors of color j is equal to aij. The matrix A is called the parameter matrix of a perfect coloring. When m = 2, we denote the two colors by W and B representing white and black respectively.

The cuboctahedral and chvatal graphs given as follow.

a4 a1 a4 a1 a5 a5 a6

a12 a9 a12 a7 a8 a6 a11 a10 a11 a8

a7 a10 a9 a3 a2 a3 a2 cuboctahedral graph chvatal graph

In this article, we enumerate parameter matrices perfect 3- coloring of the cuboctahedral and chvatal 4 regular graph. We present all possible parameter matrices for perfect 3- colorings 4 regular graph. In Section 2, we also prove which of matrices can have 3- perfect coloring in cuboctahedral graph. In Section 3, we prove 3- perfect coloring in chvatal graph. According to the paper [8], the only possible parameter matrices for the 4- regular graphs are as[ follows:] [ ] [ ] [ ] [ ] [ ] 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 A1 = 0 0 4 , A2 = 0 0 4 , A3 = 0 0 4 , A4 = 0 0 4 , A5 = 0 1 3 , A6 = 0 1 3 , [ 1 1 2] [ 1 2] 1 [ 1] 3 0 [ 2] 2 0 [ 1] 1 2 1 2 1 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 A7 = 0 1 3 , A8 = 0 1 3 , A9 = 0 2 2 , A10 = 0 2 2 , A11 = 0 2 2 , [1 3 0 ] [2 2 0 ] 1[ 1 2 ] 1[ 2 1 ] 2 1[ 1 ] [ ] 0 0 4 0 0 4 0 0 4 0 0 4 0 1 3 0 1 3 A12 = 0 2 2 ,A13 = 0 3 1 , A14 = 0 3 1 , A15 = 0 3 1 , A16 = 1 0 3 , A17 = 1 0 3 , [ 2 2 0 ] [ 1 1 2 ] [ 2 1 1 ] [ 3 1 0 ] [ 1 1 2 ] [ 2 2 0 ] 0 1 3 0 1 3 0 1 3 0 1 3 0 2 2 0 2 2 A18 = 1 2 1 ,A19 = 1 3 0 , A20 = 1 3 0 , A21 = 1 3 0 , A22 = 1 0 3 , A23 = 1 1 2 , [ 3 1 0 ] [1 0 3 ] [2 0 2 ] [3 0 1 ] [ 1 3 0 ] [ 1 2 1 ] 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 A24 = 1 2 1 , A25 = 1 3 0 , A26 = 2 0 2 , A27 = 2 0 2 , A28 = 2 1 1 , A29 = 2 2 0 , [ 1 1 2 ] [ 1 0 3 ] [ 1 1 2 ] [ 2 2 0 ] [ 2 1 1 ] [ 1 0 3 ] 0 2 2 0 4 0 0 4 0 0 4 0 0 4 0 1 0 3 A30 = 2 2 0 , A31 = 1 0 3 , A32 = 1 0 3 , A33 = 1 1 2 , A34 = 2 0 2 , A35 = 0 0 4 , [ 2 0 2 ] [ 0 1 3 ] [ 0 2 2 ] [ 0 1 3 ] [ 0 1 3 ] [ 1 2 1 ] 1 0 3 1 0 3 1 0 3 1 0 3 1 0 3 1 0 3 A36 = 0 0 4 , A37 = 0 1 3 , A38 = 0 1 3 , A39 = 0 1 3 , A40 = 0 1 3 , A41 = 0 2 2 , [ 1 3 0 ] [ 1 1 2 ] [ 1 2 1 ] [ 1 3 0 ] [ 2 2 0 ] [ 1 1 2 ] 1 0 3 1 0 3 1 0 3 1 0 3 1 1 2 1 1 2 A42 = 0 2 2 , A43 = 0 2 2 , A44 = 0 3 1 , A45 = 0 3 1 , A46 = 1 1 2 , A47 = 1 2 1 , [ 1 2 1 ] [ 2 2 0 ] [ 1 1 2 ] [ 2 1 1 ] [ 1 1 2 ] [ 2 1 1 ] 1 1 2 1 1 2 1 3 0 1 3 0 1 3 0 1 3 0 A48 = 1 3 0 , A49 = 1 3 0 , A50 = 1 0 3 , A51 = 1 0 3 , A52 = 1 1 2 , A53 = 2 0 2 , [ 1 0 3 ] [ 2 0 2 ] [ 0 1 3 ] [ 0 2 2 ] [ 0 1 3 ] [ 0 1 3 ] 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 A54 = 0 0 4 , A55 = 0 1 3 , A56 = 0 1 3 , A57 = 0 2 2 , A58 = 0 2 2 , A59 = 0 3 1 , [ 1 3 0 ] [ 1 2 1] [ 1 3 0] [ 1 1 2] [ 1 2 1] 1 1 2 2 1 1 2 1 1 2 2 0 2 2 0 2 2 0 A60 = 1 2 1 , A61 = 1 3 0 , A62 = 1 0 3 , A63 = 1 0 3 , A64 = 1 1 2 . 1 1 2 1 0 3 0 1 3 0 2 2 0 1 3

2 [ ] a b c In this paper, we generally show a parameter matrix by A = d e f , and use the fol- g h i lowing Theorems 1.3, and 1.4 see [9], and also 1.5, ..., 1.8 see [7] to obtain the parameter matrices of Perfect 3- colorings our graphs.

Theorem 1.3. Suppose that G is a k- regular graph and T is a perfect m- coloring with matrix A = (aij) in graph G. Then the sum of each row in matrix A is k.

Theorem 1.4. If T is a perfect coloring of the graph G with m colors, then any eigenvalue of the parameter matrix is an eigenvalue subset of the adjacent matrix G. [ ] a b c Theorem 1.5. Suppose that T is a perfect 3- coloring with matrix d e f , in the connected g h i graph G. Then in this case, none of the following situations will occur. (1) b = c = 0, (2) d = f = 0, (3) g = h = 0, (4) b = 0 ←→ d = 0, c = 0 ←→ g = 0, h = 0 ←→ f = 0.

Proof. The first part is proved by the connected graph and the second part isobvious. [ ] a b c Theorem 1.6. If graph G has perfect 3- coloring with matrix d e f , then it has perfect g h i 3- coloring with the following matrices:       a c b e d f e f d i h g i g h           A1 = g i h, A2 = b a c, A3 = h i g, A4 = f e d, A5 = c a b . d f e h g i b c a c b d f d e [ ] a b c Theorem 1.7. Let T be a perfect 3- coloring of a graph G with the matrix A = d e f . g h i 1. If b, c, f ≠ 0, then | | | | | | |W | = V (G) , |B| = V (G) , |R| = V (G) , b +1+ c d f h +1+ g d g b +1+ h f c 2. If b = 0, then | | | | | | |W | = V (G) , |B| = V (G) , |R| = V (G) , c +1+ ch f fg h +1+ g g fg h +1+ ch f c 3. If c = 0, then | | | | | | |W | = V (G) , |B| = V (G) , |R| = V (G) , b bf d f h +1+ dh d +1+ dh b +1+ h f bf 4. If f = 0, then | | |V (G)| | | |V (G)| | | |V (G)| W = b c , B = d cd , R = g bd . d +1+ g b +1+ bg c +1+ cd [ ] a b c Theorem 1.8. Let A = d e f be a parameter matrix of a cubic graph. Then the g h i √ tr(A)−K ± tr(A)−K 2 − det(A) eigenvalues of A are λ1,2 = 2 ( 2 ) k , λ3 = k.

3 Proof. By using the condition a + b + c = d + e + f = g + h + i = k, it is clear that one of the eigenvalues is k. Therefore det(A) = kλ1λ2. From λ2 = tr(A) − λ1 − k, we − − − 2 − get det(A) = kλ1(tr(A) λ1 k) = kλ1 + k(tr(A) √k)λ1. By solving the equation 2 − det(A) tr(A)−K ± tr(A)−K 2 − det(A) λ + (k tr(A))λ + k = 0. Obtain λ1,2 = 2 ( 2 ) k .

2 perfect 3- coloring of the cuboctahedral graph

In this section, with the help of the given theorems, we remove the parameter matrices that are not perfect coloring and examine the perfect 3- coloring of the cuboctahedral graph.

Theorem 2.1. The cuboctahedral graph has a perfect 3- coloring with the matrix A1.

Proof. By using the Theorem 1.4 the matrix A1 can be a parameter matrix because it does have the condition of Theorem 1.4 with consideration eigenvalues adjacent matrix graph and using Theorems 1.8, 1.4.

The cuboctahedral graph has perfect 3- coloring with the matrix A1. Using the Theorem 1.7 we have |W | = |B| = 2 and |R| = 8 so consider mapping T as follows:

T (a6) = T (a8) = W ,

T (a5) = T (a7) = B,

T (a1) = T (a2) = T (a3) = T (a4) = T (a9) = T (a10) = T (a11) = T (a12) = R.

It is clear that T is a perfect 3- coloring with the matrix A1.

Theorem 2.2. The cuboctahedral graph has a perfect 3- coloring with the matrix A27,A28, and A30.

Proof. By using the Theorem 1.4 the matrix A27 can be a parameter matrix because it does have the condition of Theorem 1.4 with consideration eigenvalues adjacent matrix graph and using Theorems 1.8, 1.4.

The cuboctahedral graph has perfect 3- coloring with the matrix A27. Using the Theorem 1.7 we have |W | = |B| = |R| = 4 so consider mapping T as follows:

T (a1) = T (a3) = T (a9) = T (a11) = W ,

T (a5) = T (a6) = T (a7) = T (a8) = B,

T (a2) = T (a4) = T (a10) = T (a12) = R.

It is clear that T is a perfect 3- coloring with the matrix A28.

The cuboctahedral graph has perfect 3- coloring with the matrix A28.

4 Using the Theorem 1.7 we have |W | = |B| = |R| = 4 so consider mapping T as follows:

T (a5) = T (a6) = T (a7) = T (a8) = W ,

T (a3) = T (a2) = T (a9) = T (a12) = B,

T (a1) = T (a4) = T (a10) = T (a11) = R.

It is clear that T is a perfect 3- coloring with the matrix A28.

The cuboctahedral graph has perfect 3- coloring with the matrix A30. Using the Theorem 1.7 we have |W | = |B| = |R| = 4 so consider mapping T as follows:

T (a2) = T (a4) = T (a10) = T (a12) = W ,

T (a5) = T (a6) = T (a1) = T (a9) = B,

T (a3) = T (a7) = T (a8) = T (a11) = R.

It is clear that T is a perfect 3- coloring with the matrix A30.

Theorem 2.3. There are no perfect 3- coloring of the cuboctahedral with the matrices

A37,A46, and A63.

Proof. A parameter matrix corresponding to perfect 3- coloring of the cuboctahedral may be one of the matrices A37, A46, and A63. Using the Theorem 1.4 the matrix A37 cannot be a parameter matrix, because the number of white colors is not an integer which is a contradiction with Theorem 1.7. For matrix A46 we have |W | = |B| = 3, and |R| = 6 so using this number and considering entry the matrix A46 , we being to color the vertex and examine different states. Now have the following possibilities:

(1)T (a1) = T (a5) = W and T (a6) = T (a2) = R then T (a4) = B, which is a contradiction with the second row of A46.

(2)T (a1) = T (a2) = T (a4) = R and T (a5) = W, T (a6) = B so T (a9) = W , which is a contradiction with the first row of A46. Hence cuboctahedral has no perfect 2- coloring with the matrix A46.

Also by using Theorems 1.7 for matrix A63 we have |W | = 2, |B| = 4, |R| = 6.

Since the |W | = 2 so is a contradiction with the first row of A63. Hence cuboctahedral has no perfect 3- coloring with the matrix A63.

Theorem 2.4. The cuboctahedral graph has a perfect 3- coloring with the matrix A57.

Proof. By using the Theorem 1.4 the matrix A57 can be a parameter matrix because it does have the condition of Theorem 1.4 with consideration eigenvalues adjacent matrix graph and using Theorems 1.4, 1.8.

The cuboctahedral graph has perfect 3- coloring with the matrix A57.

5 Using the Theorem 1.7 we have |W | = |B| = 3 and |R| = 6 so consider mapping T as follows:

T (a1) = T (a2) = T (a6) = W ,

T (a8) = T (a11) = T (a12) = B,

T (a3) = T (a4) = T (a5) = T (a9) = T (a10) = T (a7) = R.

It is clear that T is a perfect 3- coloring with the matrix A57.

3 Perfect 3- colorings the chvatal graph

In this section, with the help of the given theorems, we remove the parameter matrices that are not perfect coloring and examine the perfect 3- coloring of the chvatal graph.

Theorem 3.1. The chvatal graph has a perfect 3- coloring with the matrix A47.

Proof. By using the Theorem 1.4 the matrix A47 can be a parameter matrix because it does have the condition of Theorem 1.4 with consideration eigenvalues adjacent matrix graph and using Theorems 1.8, 1.4.

The chvatal graph has perfect 3- coloring with the matrix A47. Using the Theorem 1.7 we have |W | = |B| = |R| = 4 so consider mapping T as follows:

T (a7) = T (a8) = T (a11) = T (a12) = W ,

T (a1) = T (a2) = T (a3) = T (a4) = B,

T (a5) = T (a6) = T (a9) = T (a10) = R, It is clear that T is a perfect 3- coloring with the matrix A47.

Theorem 3.2. The chvatal graph has a perfect 3- coloring with the matrix A60.

Proof. By using the Theorem 1.4 the matrix A60 can be a parameter matrix because it does have the condition of Theorem 1.4 with consideration eigenvalues adjacent matrix graph and using Theorems 1.8, 1.4.

The chvatal graph has perfect 3- coloring with the matrix A60. Using the Theorem 1.7 we have |W | = |B| = |R| = 4 so consider mapping T as follows:

T (a1) = T (a2) = T (a3) = T (a4) = W ,

T (a5) = T (a6) = T (a9) = T (a10) = B,

T (a7) = T (a8) = T (a11) = T (a12) = R, It is clear that T is a perfect 3- coloring with the matrix A60.

Theorem 3.3. There are no perfect 3- coloring of the chvatal with the matrices A2,A11,A16,A17,

A18,A22,A23,A24,A39,A40,A43, and A46.

6 Proof. Using Theorem 1.4 parameter matrix corresponding to perfect 3- coloring of the chvatal may be one of thematrices A2,A11,A16,A17,A18,A22,A23,A24,A39,A40,A43,A46, and A47. In them matrices A2,A16,A17,A18,A22,A23,A24,A39,A40,and A43 cannot be a parameter matrix, because of the Theorem 1.7 the number of white colors is not an integer. Also by using Theorems 1.7 for matrix A11 we have |W | = |B| = 3,and |R| = 6 but according to the entry matrix A11, each B must be only adjacent to two B, and given the number of B and the shape of the graph, this is impossible.

Using Theorems 1.7 for matrix A46 we have |W | = |B| = 3,and |R| = 6. but according to the entry matrix A46, each W must be only adjacent to one W , and given the number of W and the shape of the graph, this is impossible. Hence chvatal has no perfect 3- coloring with the matrix A46.

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