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;j21;:::;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;j21;:::;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 cuboctahe- dral 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 ash follows:i h i h i h i h i h i 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 , h 1 1 2i h 1 2i 1 h 1i 3 0 h 2i 2 0 h 1i 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 , h1 3 0 i h2 2 0 i 1h 1 2 i 1h 2 1 i 2 1h 1 i h i 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 , h 2 2 0 i h 1 1 2 i h 2 1 1 i h 3 1 0 i h 1 1 2 i h 2 2 0 i 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 , h 3 1 0 i h1 0 3 i h2 0 2 i h3 0 1 i h 1 3 0 i h 1 2 1 i 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 , h 1 1 2 i h 1 0 3 i h 1 1 2 i h 2 2 0 i h 2 1 1 i h 1 0 3 i 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 , h 2 0 2 i h 0 1 3 i h 0 2 2 i h 0 1 3 i h 0 1 3 i h 1 2 1 i 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 , h 1 3 0 i h 1 1 2 i h 1 2 1 i h 1 3 0 i h 2 2 0 i h 1 1 2 i 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 , h 1 2 1 i h 2 2 0 i h 1 1 2 i h 2 1 1 i h 1 1 2 i h 2 1 1 i 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 , h 1 0 3 i h 2 0 2 i h 0 1 3 i h 0 2 2 i h 0 1 3 i h 0 1 3 i 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 , h 1 3 0 i h 1 2 1i h 1 3 0i h 1 1 2i h 1 2 1i 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 h i 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. h i 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. h i 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- coloring2 with3 the following2 matrices:3 2 3 2 3 2 3 a c b e d f e f d i h g i g h 6 7 6 7 6 7 6 7 6 7 A1 = 4g i h5, A2 = 4b a c5, A3 = 4h i g5, A4 = 4f e d5, A5 = 4c a b5 : d f e h g i b c a c b d f d e h i 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 =6 0, then j j j j j j jW j = V (G) , jBj = V (G) , jRj = V (G) , b +1+ c d f h +1+ g d g b +1+ h f c 2. If b = 0, then j j j j j j jW j = V (G) , jBj = V (G) , jRj = V (G) , c +1+ ch f fg h +1+ g g fg h +1+ ch f c 3. If c = 0, then j j j j j j jW j = V (G) , jBj = V (G) , jRj = V (G) , b bf d f h +1+ dh d +1+ dh b +1+ h f bf 4.