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#Post#: 9--------------------------------------------------
knight tours on special boards
By: gsgs Date: March 15, 2023, 8:11 am
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An (1, 2, 2)-Closed Knight's Tour on the 3r x 4s x 4t
Chessboard, where r≥ 1 and s, t≥ 2
S Singhun, A Sinna, P Yensuang - Mathematical Journal by The …,
2020 - ph02.tci-thaijo.org
… Then, a closed (a, b, c)-knight’s tour is a Hamiltonian cycle,
a cycle that passed through each
… Connect each path Px, we obtain a Hamiltonian cycle or a
closed (1, 2, 2)-knight’s tour C*. …
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[PDF] ysu.am
Closed knight's tour problem on some -rectangular tubes
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the m×n chessboard with the middle part missing and
the rim contains r rows and r columns. Another way is to stack k
copies
of the m × n chessboard to construct an m × n × k rectangular
chessboard
and the closed knight’s tour can be on the surface or within the
m × n × k
rectangular chessboard (see [1] and [2]). We combines these two
ideas by
stacking n copies of m×n ringboard of width r, which we call the
(m,n,k,r)-
rectangular tube and each stacking copy is called the level of
(m,n,k,t)-tube.
In this talk, we show an algorithm for a closed knight’s tour
for (3,3,k,1)-
rectangular tube and give a new algorithm for a closed knight’s
tour for
(4,4,k,1)-rectangular tube which is shorter than [3]. We show
the sufficient
and necessary conditions for (3,n,k,1)-tube when n 6= 5 and
(5,5,k,1)-tube
when k 6= 5. Moreover, closed knight’s tours for (3,5,k,1)-tube
when k ≡ 0
(mod 4) and (m,m,k,1)-tube when m(> 5) is odd and k ≡ 0
(mod 4) are
obtained.
[1] J. DeMaio, Which Chessboards have a Closed Knight’s Tour
within the
Cube? The electronic j. of combin. (14) (2007), #R32.
[2] J. DeMaio and B. Mathew, Which Chessboards have a Closed
Knight’s
Tour within the Rectangular Prism? The electronic j. of combin.
(18)
(2011), #P8.
[3] N. Loykaew, S. Singhun and R. Boonklurb, A closed knight’s
tour prob-
lem on the (3,n,1)-tube and the (4,n,1)-tube. Proceeding of The
13th
conference of young algebraists in Thailand, 6-9 December, 2017,
Sil-
pakorn University, accepted
[4] A. L. Schwenk, Which rectangular chessboards have a knight’s
tour.
Math. Magazine (64) (1991), 325–332.
[5] H. R. Wiitala, The Knight’s Tour Problem on Boards with
Holes. Re-
search Experiences for Undergraduates Proceedings (1996),
132–151.
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