Perfect 2-coloring of the quartic graphs with order at most 8

In this paper, we study perfect 2-coloring of the quartic graphs with at most 8 vertices. The problem of the existence of perfect coloring is a generalization of the concept of completely regular codes, given by Delsarte.


Introduction
Let X = (V(X), E(X)) be a connected graph. A vertex coloring with m colors is perfect if there is a matrix M = ( ij ) i,j=1,2,…m such that the cardinal of vertices of label j connected to a specified vertex of label i is equal to i,j .
In particular, all vertices of the same color in X have the same degree. This type of coloring gives a partition of V(X) that is called equitable partitions. These partitions of vertices have two properties as follows: The first property is that the vertices of each part P i induce a regular graph, and the second is the edges between P i and P j induce a half-regular graph. In Section 3, we find the parameter matrices of a perfect 2-coloring. Equitable partitions were previously studied in [7].
In fact, the concept of a perfect m-coloring is bridge of among algebraic combinatorics, graph theory and coding theory (including 1-perfect codes). For the definition of 1-perfect codes and more details, we refer the reader to [1].
This type of coloring is generalized the concept of completely regular codes introduced by P. Delsarte ( [8]).The problem for the Johnson graphs has been studied (see [4], see [3], see [6]).
In subsequent attempts, the problem of existence of perfect coloring was examined on other graphs. For example, Fon-Der-Flass settled perfect coloring with two colors of n-dimensional hypercube graphs Q n for n < 24 (see [5]). In this paper, we will list all parameter matrices of perfect 2-coloring of the quartic graphs up to order 8.

Notations and preliminaries result on the existence of perfect coloring
In this section, some basic definitions that will be used in this paper are given and we declare some conditions for existence of perfect 2-coloring of the quartic graphs.
In this paper, all the graphs are simple. A graph where k is a nonnegative integer number. In particular, graphs that are regular of degrees 3, 4, 5 and 6 are called, respectively, cubic, quartic, quantic and sixtic.
In the following remark, we classify the quartic graphs with at most 8 vertices (see [9]).

Remark 1
The quartic graphs are divided into four classes based on their vertices. This classification is shown in Figs. 1, 2, 3 and 4. For simplicity, we name the vertices as shown in these figures.
We now introduce the concept of perfect coloring.

3
Definition 1 Let X = (V(X), E(X)) be a connected graph. A perfect m-coloring with parameter matrix M = ( ij ) i,j=1,2,…m is a map C from V(X) to the set of colors {1, 2, … , m} such that C is surjective, and for a fix vertices v where In this case, we say C is a P.m.C (or just P.C). Here, we study P.2.C, where two colors are red and blue.
In this case, we consider the parameter matrix of P.2.C generated by replacing the colors with the primary coloring is equal. This means the parameter matrix a b a ′ b ′ is equal to the parameter matrix b ′ b a ′ a . We call equality of parameter matrix. Some properties of P.C have been studied recently (see [4,6,8]). The main result of them is the next proposition that enumerates the cardinal number of blue vertices in a P.2.C with matrix M = ( ij ) i,j=1,2 (see [3]).

Then we have
Now, the following remark provides useful information about calculation P.2.C of the quartic graphs.

Remark 2
Suppose that X = (V(X), E(X)) be a simple connected graph. Then necessary conditions for existence of 1. Degree condition: This simple condition that results from the regularity of X tells us:

Connectedness condition:
Since the graph X is connected, another condition is induced as follows: b, a ′ ≠ 0.
We will define a new definition for the third condition. The number is called an eigenvalue of a graph X, if is an eigenvalue of the adjacency matrix of X. The number is called an eigenvalue of a P.2.C with the matrix M, if is an eigenvalue of M.
The next theorem gives us the third condition and the connection between two types of eigenvalues (see [4]):

Theorem 1 Let C be a P.m.C of a graph X. Thus any eigenvalues of C are an eigenvalues of X.
It has been proved that a P.2.C of a k-regular graph X has just two eigenvalues (see [4]). 2 of a regular graph X of valency k, then the numbers 11 − 21 and k are eigenvalues of C and therefore are eigenvalues of X.

Lemma 1
Given an arbitrarily connected quartic graph X, the following ten matrices are the only matrices that can be selected as acceptable parameter matrices of a perfect 2-coloring C of X: Proof By Remark 2, Theorem 1, Proposition 2 and equality of parameter matrix, proof is clear. ◻ Note that the word acceptable in the previous lemma states that some of the listed matrices may not be a parameter matrix of a P.2.C of X. Therefore, we reduce them by Proposition 1 in the next lemma (Tables 1, 2 Table 1 Acceptable parameter matrices of quartic graph with order 5 Matrices Table 2 Acceptable parameter matrices of quartic graph with order 6 Matrices

Perfect 2-coloring of the quartic graphs with order at most 8
In the previous section, we determined all acceptable parameter matrices of a P.2.C of a connected quartic graph with at most 8 vertices. In this section, we show which of them have a structure. In the other words, we introduce a perfect 2-coloring C for them.

Lemma 3
Assume that X be a connected k-regular graph.
Then there are no P.2.C with parameter matrix M = a b a � b � such that a = |B| ≤ k.
Proof Suppose, contrary to our claim, there is a P.2.C with presumed conditions. Let V be an arbitrary vertex with blue color. So, since a = k , we conclude that all k vertices adjacent to v must be blue. This is in contradiction with |B| ≤ k . ◻

Corollary 2
There are no P.2.C of Q 2 with the matrices M 9 and M 10 .
Proof From Lemma 2, we conclude that graph Q 2 has the conditions of Lemma 3, So the proof is straightforward. ◻ We will discuss about acceptability of matrix M 7 is the following lemma: Lemma 4 There are no P.2.C of graphs Q 6 , Q 7 and Q 8 with matrix M 7 .
Proof Suppose that there is a P.2.C of the graphs with M 7 . Note that both adjacent vertices of the graph Q 6 , Q 7 and Q 8 have a common vertex. On the other hand, from 11 = 1 in M 7 , we obtain that there are two adjacent vertices with blue color. So, these two blue vertices have common red vertex. This is a contradiction with 21 = 1 . ◻

Lemma 5
There are no P.2.C of graphs Q 5 , Q 6 , Q 7 , Q 8 and Q 9 with the matrices M 1 and M 10 .

Proof The proof of this lemma is similar to the proof of Lemma 4. ◻
In the following, we will state the main result of this article. For this, we determine which class of the quartic graphs have a structure. Proof By Corollary 1, only matrices M 4 and M 6 are acceptable as parameter matrices of a P.2.C of Q 1 . Two mapping C 1 and C 2 determine two structures for Q 1 with matrices M 4 and M 6 , respectively, as follows: It is clear that C 1 and C 2 are P.
It is obvious that the function C 1 and C 2 stats a P.2.C for Q 2 with matrices M 3 and M 8 , respectively. In the following, we prove that other matrices can not be as parameter matrices of Q 2 . First, we show that there is no P.2.C with the matrix M 1 for the graph Q 2 . Contrary to our claim, suppose that C is a P.2.C of Q 2 with the matrix M 1 . There is no loss of generality in assuming C( 1 ) = 1 . From M 1 , we conclude that C( 2 ) = C( 3 ) = C( 4 ) = C( 5 ) = 2 . Hence only one vertex remains, even if it is blue, in contradiction with the number of blue vertices (|B| = 3).
We have shown in Corollary 2 that there is no P.2.C of Q 2 with the matrices M 9 and M 10 . Now we prove that there is no P.2.C of Q 2 with the matrix M 5 . Suppose that there is a P.2.C of Q 2 with the matrix M 5 , say C. From 11 = 1 , without loss of generality, we assume that C( 1 ) = C( 2 ) = 1 . Thus, from 12 = 3 , we have C( 3 ) = C( 4 ) = C( 5 ) = C( 6 ) = 2 . In this case, the red vertex 6 is connected to the other three red vertices, which leads to a contradiction. ◻

Theorem 4 Only the graph Q 4 has a P.2.C between the quartic graphs with 7 vertices.
Proof According to Corollary 1, we see that only parameter matrix of Q 3 and Q 4 is M 2 . First, we give a structure for Q 4 with matrix M 2 by defining the function C as follows: It is easy to check that C is a P.2.C of Q 4 with matrix M 2 . Now we prove that there is no P.2.C with the matrix M 2 for the graph Q 3 . Let C be a perfect 2-coloring of Q 3 with the matrix M 2 . Let C( 1 ) = 1 . From 12 = 4 , we deduce that all vertices are red except for adjacent vertices 4 and 5 . On the other hand, we know that |B| = 3 . Thus C( 4 ) = C( 5 ) = 1 . This is a contradiction with 11 = 0 . ◻ Finally, we get the following theorem ( which is a contradiction with 22 = 1 . The same conclusion can be drawn for M 8 . For graph Q 7 , we only need to show that there is no P.2.C of Q 7 with matrix M 5 . Suppose there is a P.2.C of Q 7 with matrix M 5 . Without loss of generality, we can suppose that C( 1 ) = C( 2 ) = 1 . Since 12 = 3 , we have C( 6 ) = C( 7 ) = C( 8 ) = 2 , contrary to 22 = 1 . It remains only to prove that there is no P.2.C of Q 10 with matrix M 5 . On the contrary, suppose that there is a P.2.C of Q 10 with matrix M 5 . From 11 = 1 , we conclude that the two adjacent vertices in Q 10 , all other vertices will be red. This gets to a contradiction with |B| = 4, and the proof is complete. ◻

Corollary 3
The only P.2.C of the quartic graphs with at most 8 vertices is prefect coloring with listed matrices in the following table: (Table 7).

Conclusion
The perfect coloring of graphs is closely related to coding theory, algebraic theory, graph theory and combinatorics, including designs. We can consider perfect m-coloring as a generalization of the concept of completely regular codes presented by P. Delsarte for the first time. This class of codes has been of interest to coding theorists and graph theorists alike. In this way, the problem of existence of perfect coloring of the Johnson graphs, the generalized Petersen graphs and the n-dimensional hypercube graphs Q n for n < 24 has been settled in [2][3][4]6] and [5].
In this paper, we have listed all parameters of existing perfect 2-coloring in the quartic graphs with order at most 8. The question of existence of perfect 3-coloring of the quartic graphs remains open.
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