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Silver cubes

Ghebleh, M ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1007/s00373-008-0799-3
  3. Publisher: 2008
  4. Abstract:
  5. An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n - 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K n d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K n , and c : V(K n d)→ {1,2,...,d(n-1)+1} is a vertex colouring where, for ν ∈ I, the closed neighbourhood N[ν] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory. © 2008 Springer Japan
  6. Keywords:
  7. Silver matrix ; Graph colouring ; Coding theory ; Domination in graphs ; Finite projective geometry
  8. Source: Graphs and Combinatorics ; Volume 24, Issue 5 , 2008 , Pages 429-442 ; 09110119 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s00373-008-0799-3