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Article

Closed Knight’s Tours on (m,n,r)-Ringboards

1
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2
Department of Mathematics, Ramkhamhaeng University, Bangkok 10240, Thailand
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(8), 1217; https://doi.org/10.3390/sym12081217
Received: 2 July 2020 / Revised: 21 July 2020 / Accepted: 23 July 2020 / Published: 25 July 2020
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
A (legal) knight’s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight’s tour is a knight’s move that visits every square on a given chessboard exactly once and returns to its start square. A closed knight’s tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n>2r, an (m,n,r)-ringboard or (m,n,r)-annulus-board is defined to be an m×n chessboard with the middle part missing and the rim contains r rows and r columns. In this paper, we obtain that a (m,n,r)-ringboard with m,n3 and m,n>2r has a closed knight’s tour if and only if (a) m=n=3 and r=1 or (b) m,n7 and r3. If a closed knight’s tour on an (m,n,r)-ringboard exists, then it has symmetries along two diagonals. View Full-Text
Keywords: legal knight’s move; closed knight’s tour; open knight’s tour; Hamiltonian cycle; ringboard; annulus-board legal knight’s move; closed knight’s tour; open knight’s tour; Hamiltonian cycle; ringboard; annulus-board
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MDPI and ACS Style

Srichote, W.; Boonklurb, R.; Singhun, S. Closed Knight’s Tours on (m,n,r)-Ringboards. Symmetry 2020, 12, 1217. https://doi.org/10.3390/sym12081217

AMA Style

Srichote W, Boonklurb R, Singhun S. Closed Knight’s Tours on (m,n,r)-Ringboards. Symmetry. 2020; 12(8):1217. https://doi.org/10.3390/sym12081217

Chicago/Turabian Style

Srichote, Wasupol, Ratinan Boonklurb, and Sirirat Singhun. 2020. "Closed Knight’s Tours on (m,n,r)-Ringboards" Symmetry 12, no. 8: 1217. https://doi.org/10.3390/sym12081217

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