Intersection Arrays of Completely Regular Codes of Covering Radius One in Generalized Petersen Graphs

Abstract

We determine all possible intersection arrays of completely regular codes of covering radius one in the generalized Petersen graphs \(GP(n,k)\), where \(n\geq 3\) and \(1\leq k<n/2\). In the equivalent language of perfect colorings, this amounts to enumerating all quotient matrices of perfect \(2\)-colorings, up to interchanging the two colors. Since \(GP(n,k)\) is cubic, there are only six possible nontrivial quotient matrices. For each of them, we give necessary and sufficient arithmetic conditions on \(n\) and \(k\) for its existence. The feasible cases are realized by explicit periodic colorings. The nonexistence part is obtained by reducing the local coloring conditions to cyclic systems of linear equations and applying a Fourier argument on roots of unity. Together with the previously known cases \(GP(n,2)\) and \(GP(n,3)\), the results give a complete arithmetic classification of quotient matrices, and hence of covering-radius-one completely regular code parameters, in the generalized Petersen family.