Dominaion on he m n Toroidal Chessboard by ooks and ishops Joe DeMaio William P. Faus Deparmen of Mahemaics and Saisics
Dominaion The dominaion number,, ofa chessboard is defined as he minimum number of pieces ha are needed o hreaen or occupy every square on a chess board. This is an example of a dominaed chessboard wih rooks. Noice ha he movemen of he rooks allow hem o dominae any square wihin heir row. (Similarly hey can dominae any square wihin heir column.)
Toal Dominaion The oal dominaion number,,ofa chessboard is defined as he minimum number of pieces ha are needed o hreaen every square on a chess board. For oal dominaion a piece no longer hreaens he square i occupies. I is immediae ha 2. A oal dominaing se is also a dominaing se. For every verex in a dominaing se, a neighbor can be included o consruc a oal dominaing se. The rooks dominae he unoccupied squares as well as each oher wih heir verical movemen.
The Torus
ooks on he Torus On he orus, he rook hreaens no addiional squares. Thus, for m, n 2, m,n m,n m,n m,n min m, n.
Addiional Freedom on he Torus for ishops 1 3 2 1 4 5 6 7 8 9 2 4 5 3 6 7 8 9 The figure above demonsraes how he movemens of he bishop chesspiece on a wo-dimensional chessboard can be ransformed ino a graph wih verices represening a bishop and each edge represening he abiliy o move o a differen verex. 1 3 2 1 4 5 6 7 8 9 2 4 5 3 6 7 8 9 The figure above demonsraes how he movemens of he bishop chess piece on he hree-dimensional oroidal chessboard can be ransformed ino a graph. Noice ha here are many more edges beween he verices compared o he graph of he wo-dimensional chessboard.
ishops Movemen on he Torus Since he orus has no borders, a bishop will always complee a cycle and reurn o is saring posiion from he opposie direcion. 1 2 3 8 4 13 9 5 15 6 7 10 11 12 14 21 22 23 24 25 16 17 18 19 20 3 2 1 7 6 12 11 4 5 8 9 10 13 14 15 17 16 18 19 20 22 21 23 24 25 S-diagonals D-diagonals Theorem 1: ishops moves are monochromaic if and only if boh m and n are even. Theorem 2: On he m n recangular orus, a bishop will aack mn lcm m, n squares on eiher he gcd m,n s-diagonal or d-diagonal.
ishop s Dominaion Number on he Torus Theorem 3: For he recangular m n orus, gcd m, n 1 if and only if m,n 1. Theorem 4: For he square orus of odd side n, n n. For odd n i is easy o redraw he bishops graph in he wo-dimensional plane o resemble he rooks graph jus as one does o analyze n. 1 2 3 4 5 1 6 2 14 10 1 22 18 6 7 8 9 10 11 7 3 16 12 8 4 20 11 7 3 24 11 12 13 14 15 21 17 13 9 5 22 18 14 10 21 17 13 9 5 16 17 18 19 20 23 19 15 24 20 2 23 19 15 6 21 22 23 24 25 25 8 4 25 16 12 Quadran board formaion
Theorem 5: For he square orus of even side n, n n. For square boards of even side n i is no possible o redraw he bishop s graph in he wo-dimensional plane o resemble he rook s graph. Theorem 6: For he recangular m km orus, m,km m. Theorem 7: For he recangular jm km orus, jm,km m if and only if gcd j, k 1.
When working wih bishops on he recangular m n orus, i is sufficien o focus on he iniial square of side gcd m, n. y combining he previous heorems ogeher we achieve our main resul. Theorem 8: For he recangular m n orus, m,n gcd m, n.
ishop s Toal Dominaion Number on he Torus On he square orus of odd side n 3, n n. This is apparen since all bishops may be placed along he main diagonal of he board. On he square orus of even side n 4, n n. A differen arrangemen of bishops is needed since bishops are locked o squares of one color. In he even case i is necessary o place bishops on n 2 squares of boh he main diagonal and he minor diagonal.
Consequences for he Queen s Dominaion Number on he Torus Corollary 1: For he recangular m n orus, if gcd m, n 1 hen Q m,n 1. Corollary 2: For he recangular m n orus, if gcd m, n 2 and n 4 hen Q m,n 2.
Fuure Work The following able gives he known and unknown values for graphs on boh he wo-dimensional chessboard and he oroidal chessboard: Graph G ir G G i G 0 G Γ G I G n n n n n n 2n 4 n n n n 2n 2 2n 2 4n 14 m,n? min m, n min m, n min m, n?? m,n?????? m,n? min m, n min m, n min m, n?? m,n? gcd m, n gcd m, n gcd m, n?? Table 1: Dominaion Chain Values
eferences A. P. urger, C. M. Mynhard, and W. D. Weakley, Queen s Graph for Chessboards on he Torus, Ausralian Journal of Combinaorics, Volume 24 (2001), pp 231-246. A. P. urger, C. M. Mynhard, and W. D. Weakley, The Dominaion Number of he Toroidal Queen s Graph Of Size 3k 3k, Ausralian Journal of Combinaorics, Volume 28 (2003), pp 137-148. G. Carnes, Queens on Non-square Tori, The Elecronic Journal of Mahemaics, 8 (2001), #N6. E. J. Cockayne, Chessboard Dominaion Problems, Discree Mah, Volume 86 (1990), pp 13-20. E. J. Cockayne,. Gamble, and. Shepard, Dominaion Parameers for he ishops Graph, Discree Mah, Volume 58 (1986), pp 221-227. G. H. Fricke, S. M. Hedeniemi, A. A. Mcae, C. K. Wallis, M. S. Jacobson, W. W. Marin, and W. D. Weakly, Combinaorial Problems on Chessboards: A rief Survey, Graph Theory, Combinaorics and Applicaions, Volume 1 (1995), pp 507-528. T. W. Haynes, M. A. Henning, Dominaion in Graphs, Handbook of Graph Theory, CC Press, oca aon,
2004. S. M. Hedeniemi, S. T. Hedeniemi, and. eynolds, Combinaorial Problems on Chessboards: II, Dominaion in Graphs: Advanced Topics, Marcel Dekker, Inc., New York, 1998. J. J. Wakins, Across he oard: The Mahemaics of Chessboard Problems, Princeon Universiy Press, Princeon and Oxford, 2004. E. W. Weissein, Torus, MahWorld-A Wolfram Web esource. hp://mahworld.wolfram.com/torus.hml A. M. Yaglom and I. M. Yaglom, Challenging Mahemaical Problems wih Elemenary Soluions, Holden-Day, Inc., San Francisco, 1964.