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Perfect 3-colorings of Heawood graph | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 55، دوره 12، شماره 1، مرداد 2021، صفحه 713-717 اصل مقاله (129.86 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/IJNAA.2021.4901 | ||
| نویسنده | ||
| Mehdi Alaeiyan | ||
| School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846, Iran | ||
| تاریخ دریافت: 16 مهر 1397، تاریخ بازنگری: 12 خرداد 1398، تاریخ پذیرش: 17 خرداد 1398 | ||
| چکیده | ||
| Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts $A_1, . . . , A_m$ such that, for all $i, j \in \{1, . . . , m\}$, every vertex of $A_i$ is adjacent to the same number of vertices, namely, $a_{ij}$ vertices, of $A_j$. The matrix $A = (a_{ij} )$, $i, j \in \{1, 2,... , m\}$, is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the Heawood graph. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the Haywood graphs. | ||
| کلیدواژهها | ||
| perfect coloring؛ parameter matrices؛ cubic graph | ||
| مراجع | ||
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[1] M. Alaeiyan and A.A. Abedi, Perfect 2-colorings of Johnson graphs J(4, 3), J(4, 3), J(6,3) and Petersen graph, Ars Combinatorial, 2018. [2] M. Alaeiyan and H. Karami, Perfect 2-colorings of the generalized Petersen graph, Proc. Math. Sci. 126 (2016) 1-6. [3] M. Alaeiyan and A. Mehrabani, Perfect 3-colorings of cubic graphs of order 10, Electronic J. Graph Theory Appl., to appear. [4] S.V. Avgustinovich and I. Yu. Mogilnykh, Perfect 2-Colorings of Johnson Graphs J(6, 3) and J(7, 3), Lecture Notes in Computer Science, 2008. [5] S.V. Avgustinovich and I.Yu. Mogilnykh, Perfect colorings of the Johnson graphs J(8, 3) and J(8, 4) with two colors, J. Appl. Ind. Math. 5 (2011) 19-30. [6] D.G. Fon-Der-Flaass, A bound on correlation immunity, Siber. Electronic Math. Rep. J. 4 (2007) 133-135. [7] D.G. Fon-Der-Flaass, Perfect 2-colorings of a hypercube, Siber. Math. J. 4 (2007) 923-930. [8] D.G. Fon-Der-Flaass, Perfect 2-colorings of a 12-dimensional cube that achieve a bound of correlation immunity, Siber. Math. J. 4 (2007) 292-295. [9] A.L. Gavrilyuk and S.V. Goryainov, On perfect 2-colourings of Johnson graphs J(v,3), J. Combin. Designs 21 (2013) 232-252. [10] C. Godsil and R. Gordon, Algebraic Graph Theory, Springer Science+Business Media, LLC, 2004. [11] C.Godsil, Compact graphs and equitable partitions, Linear Alg. Appl. 255(1-3) (1997) 259-266. | ||
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