J. Indones. Math. Soc. Vol. 24, No. 02 (2018), pp. 47–53. PERFECT 3-COLORINGS OF GP(5,2), GP(6,2), AND GP(7,2) GRAPHS Mehdi Alaeiyan 1 , Hamed Karami 2 , and Sajjad Siasat 3 1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846, Iran alaeiyan@iust.ac.ir 2 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846, Iran h karami@iust.ac.ir 3 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846, Iran s siasat@depmath.iust.ac.ir Abstract. In this paper we enumerate the parameter matrices of all perfect 3- colorings of GP(5,2), GP(6,2), and GP(7,2) graphs. Key words and Phrases: perfect coloring, equitable partition, generalized Petersen graph. Abstrak. Pada paper ini ditentukan matrik parameter dari 3-pewarnaan sempurna dari graf GP(5,2), GP(6,2), dan GP(7,2). Kata kunci:pewarnaan sempurna, partisi equitable, generalisasi graf Petersen 1. INTRODUCTION The concept of a perfect m-coloring plays an important role in graph theory, algebraic combinatorics, and coding theory (completely regular codes). There is another term for this concept in the literature as ”equitable partition” (see Godsil [8]). The existence of completely regular codes in graphs is a historical problem in mathematics. Completely regular codes are a generalization of perfect codes. In 1973, Delsarte conjectured the non-existence of nontrivial perfect codes in Johnson graphs. Therefore, some effort has been done on enumerating the parameter matri- ces of some Johnson graphs, including J (6, 3), J (7, 3), J (8, 3), J (8, 4), and J (v, 3) (v odd) (see Avgustinovich and Mogilnykh [2, 3] and Gavrilyuk and Goryainov [7]). 2010 Mathematics Subject Classification: 05C15 Received: 20-02-2016, revised: 06-02-2018, accepted: 06-02-2018. 47