Characterizing r-perfect codes in direct products of two and three cycles

Jerebic, Janja; Klavžar, Sandi; Špacapan, Simon

doi:10.1016/j.ipl.2004.12.010

Cited by 11 publications

(12 citation statements)

References 15 publications

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“…This enables us to conclude that for the direct product of four cycles no other codes exist. (For two and three factors this is proved in [7].) Based on these results we conclude the paper with a conjecture that no other codes exist.…”

Section: Introductionsupporting

confidence: 53%

An almost complete description of perfect codes in direct products of cycles

Klavžar

Špacapan

Žerovnik

2006

Advances in Applied Mathematics

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Let G = × n i=1 C i be a direct product of cycles. It is proved that for any r 1, and any n 2, each connected component of G contains an r-perfect code provided that each i is a multiple of r n + (r + 1) n . On the other hand, if a code of G contains a given vertex and its canonical local vertices, then any i is a multiple of r n + (r + 1) n . It is also proved that an r-perfect code (r 2) of G is uniquely determined by n vertices, and it is conjectured that for r 2 no other codes in G exist other than the constructed ones.

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Section: Introductionsupporting

confidence: 53%

An almost complete description of perfect codes in direct products of cycles

Klavžar

Špacapan

Žerovnik

2006

Advances in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…It has been proved in [7] that (each connected component of) the direct product of two cycles contains an r-perfect code, r ≥ 1, if and only if the length of each cycle is a multiple of (r + 1) 2 + r 2 . Moreover, an r-perfect code of the direct product of two cycles is uniquely determined by two vertices (on each connected component) and these are the only perfect codes that exist.…”

Section: Tablementioning

confidence: 99%

“…Recall the construction from [7]: for a given r ≥ 1 we define s = 2r + 1 and t = (r + 1) 2 + r 2 and use this notation throughout the paper. Let P be an r-perfect code of a connected component of C m × C n and (i, j) ∈ P then…”

Section: Tablementioning

confidence: 99%

“…In [7], this set was denoted by P ij (i + 1, j + s). In the second case, the code is determined by the vertices (i, j) and (i + s, j + 1).…”

Section: Tablementioning

confidence: 99%

“…The concept has applications in game theory and frequency assignment [11,4]. Perfect codes of direct product graphs and in particular, perfect codes in products of cycles were studied recently [7][8][9][10]14]. In this paper, we study existence of perfect codes in direct graph bundles of cycles over cycles and provide a complete solution.…”

Section: Introductionmentioning

confidence: 99%

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