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International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 55 | Number 5 | Year 2018 | Article Id. IJMTT-V55P544 | DOI : https://doi.org/10.14445/22315373/IJMTT-V55P544

Dominance of Rooks and Bishops in Fractal Chessboard


R.Sangeetha, G.Jayalalitha
Abstract

This paper proves that a maximum of N - 2 non- attacking Rooks and N non- attacking Bishops can be placed in a N * N Fractal Chessboard (FC) which is defined as a board that grows progressively in a consistent manner using a 2 * 2 chessboard (B) by adding the copy of B to its sides and corners.

Keywords
Rook Domination Number, Rook Independence Separation Number, Fractal Chessboard, Nonattacking Rook, Non- attacking Bishop.
References

[1] Cockayne,E.J. & Hedetniemi,S.T., Towards a theory of domination in graphs, Networks, 7, pp. 247-261, 1977
[2] Cockayne,E.J., B.Gamble & B.Shepherd, Domination Parameters for the Bishops graph, Discrete Mathematics 58 , pp. 221-227, 1986
[3] Cockayne,E., Chessboard domination problems, Discrete Mathematics 86 ,pp. 13-20,1990
[4] De Jaenisch,C.F., Applications de I’Analyse Mathematique an Jeu des Echecs, Petrograd 1862.
[5] Douglas Chatham,R., Independence and Domination Separation on Chessboard Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 68 ,pp. 3 -17, 2008
[6] Hedetniemi, S.M., Hedetniemi, S.T., & Reynolds,R., Combinatorial problems on Chessboards. II. In Domination in Graphs 67,pp.133- 162, 1998
[7] Wayne Goddard & Michael Heming, A., Independent Domination in Graphs : A survey and recent results, Discrete Mathematics 313, pp. 839-854, 2013
[8] Zhao,K., The Combinatorics of Chessboards, Ph.D. dissertation, Mathematics, City University of Newyork ,Newyork, 1908.
[9] Doug Chatham, The Maximum Queens Problem with Pawns, Recreational Mathematics Magazine, 6, pp. 95-102, 2016.

Citation :

R.Sangeetha, G.Jayalalitha, "Dominance of Rooks and Bishops in Fractal Chessboard," International Journal of Mathematics Trends and Technology (IJMTT), vol. 55, no. 5, pp. 340-344, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V55P544

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