NDLI: Silver Cubes

Content Provider	Springer Nature Link
Author	Ghebleh, Mohammad Goddyn, Luis A. Mahmoodian, Ebadollah S. Verdian Rizi, Maryam
Copyright Year	2008
Abstract	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 $$c:V(K_n^d)\rightarrow \{1,2,\dots,d(n-1)+1\}$$ is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] 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.
Starting Page	429
Ending Page	442
Page Count	14
File Format	PDF
ISSN	09110119
Journal	Graphs and Combinatorics
Volume Number	24
Issue Number	5
e-ISSN	14355914
Language	English
Publisher	Springer Japan
Publisher Date	2008-10-26
Publisher Place	Japan
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Silver matrix Graph colouring Coding theory Domination in graphs Finite projective geometry Engineering Design Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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