Queen's graph
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8
d8 white circle
h8 white circle
a7 white circle
d7 white circle
g7 white circle
b6 white circle
d6 white circle
f6 white circle
c5 white circle
d5 white circle
e5 white circle
a4 white circle
b4 white circle
c4 white circle
d4 white queen
e4 white circle
f4 white circle
g4 white circle
h4 white circle
c3 white circle
d3 white circle
e3 white circle
b2 white circle
d2 white circle
f2 white circle
a1 white circle
d1 white circle
g1 white circle
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In an queen's graph, each square of the chessboard above is a vertex. There is an edge between any two vertices that a queen could move between; as an example, the vertices adjacent to d4 are marked with a white dot (i.e. there is an edge from d4 to each marked vertex).
Vertices
Chromatic numbern if
PropertiesBiconnected, Hamiltonian
Table of graphs and parameters

In mathematics, a queen's graph is an undirected graph that represents all legal moves of the queen—a chess piece—on a chessboard. In the graph, each vertex represents a square on a chessboard, and each edge is a legal move the queen can make, that is, a horizontal, vertical or diagonal move by any number of squares. If the chessboard has dimensions , then the induced graph is called the queen's graph.

Independent sets of the graphs correspond to placements of multiple queens where no two queens are attacking each other. They are studied in the eight queens puzzle, where eight non-attacking queens are placed on a standard chessboard. Dominating sets represent arrangements of queens where every square is attacked or occupied by a queen; five queens, but no fewer, can dominate the chessboard.

Colourings of the graphs represent ways to colour each square so that a queen cannot move between any two squares of the same colour; at least n colours are needed for an chessboard, but 9 colours are needed for the board.

Properties

There is a Hamiltonian cycle for each queen's graph, and the graphs are biconnected (they remain connected if any single vertex is removed). The special cases of the and queen's graphs are complete.[1]

Independence

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c8 white queen
e7 white queen
b6 white queen
h5 white queen
a4 white queen
g3 white queen
d2 white queen
f1 white queen
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An independent set of size 8 for an chessboard (such sets are necessarily also dominating).[2]

An independent set of the graph corresponds to a placement of several queens on a chessboard such that no two queens are attacking each other. In an chessboard, the largest independent set contains at most n vertices, as no two queens can be in the same row or column.[2] This upper bound can be achieved for all n except n=2 and n=3.[3] In the case of n=8, this is the traditional eight queens puzzle.[2]

Domination

A dominating set of the queen's graph corresponds to a placement of queens such that every square on the chessboard is either attacked or occupied by a queen. On an chessboard, five queens can dominate, and this is the minimum number possible[4]:113–114 (four queens leave at least two squares unattacked). There are 4,860 such placements of five queens, including ones where the queens control also all occupied squares, i.e. they attack respectively protect each other. In this subgroup, there are also positions where the queens occupy squares on the main diagonal only[4]:113–114 (e.g. from a1 to h8), or all on a subdiagonal (e.g. from a2 to g8).[5][6]

abcdefgh
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f6 white queen
c5 white queen
e4 white queen
g3 white queen
d2 white queen
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abcdefgh
A dominating (and independent) set of size 5.
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g7 white queen
f6 white queen
e5 white queen
c3 white queen
a1 white queen
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A dominating set on the main diagonal.
abcdefgh
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g8 white queen
e6 white queen
d5 white queen
c4 white queen
a2 white queen
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A dominating set on a sub diagonal.

Modifying the graph by replacing the non-looping rectangular chessboard with a torus or cylinder reduces the minimum dominating set size to four.[4]:139

a5 black circleb5c5 black circled5e5 black circle
a4b4 black circlec4 black circled4 black circlee4
a3 black circleb3 black circlec3 black crossd3 black circlee3 black circle
a2b2 black circlec2 black circled2 black circlee2
a1 black circleb1c1 black circled1e1 black circle
Dotted squares are adjacent to the centre square. The 8 non-adjacent squares are adjacent in the corresponding knight's graph.[4]:117

The queen's graph is dominated by the single vertex at the centre of the board. The centre vertex of the queen's graph is adjacent to all but 8 vertices: those vertices that are adjacent to the centre vertex of the knight's graph.[4]:117

Domination numbers

Define the domination number d(n) of an queen's graph to be the size of the smallest dominating set, and the diagonal domination number dd(n) to be the size of the smallest dominating set that is a subset of the long diagonal. Note that for all n. The bound is attained for , but not for .[4]:119

The domination number is linear in n, with bounds given by:[4]:119,121

Initial values of d(n), for , are 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5 (sequence A075458 in the OEIS).

Let Kn be the maximum size of a subset of such that every number has the same parity and no three numbers form an arithmetic progression (the set is "midpoint-free"). The diagonal domination number of an queen's graph is .[4]:116

Define the independent domination number ID(n) to be the size of the smallest independent, dominant set in an queen's graph. It is known that .[7]

Colouring

Refer to caption
A 9-colouring of the queen's graph.[8] Notice that each pair of squares with the same colour are not on the same rank, file or diagonal, so a queen could not move directly between the squares.

A colouring of the queen's graph is an assignment of colours to each vertex such that no two adjacent vertices are given the same colour. For instance, if a8 is coloured red then no other square on the a-file, eighth rank or long diagonal can be coloured red, as a queen can move from a8 to any of these squares. The chromatic number of the graph is the smallest number of colours that can be used to colour it.

In the case of an queen's graph, at least n colours are required, as each square in a rank or file needs a different colour (i.e. the rows and columns are cliques).[1] The chromatic number is exactly n if (i.e. n is one more or one less than a multiple of 6).[9]

The chromatic number of an queen's graph is 9.[10]

Irredundance

A set of vertices is irredundant if removing any vertex from the set changes the neighbourhood of the set i.e. for each vertex, there is an adjacent vertex that is not adjacent to any other vertex in the set. This corresponds to a set of queens which each uniquely control at least one square. The maximum size IR(n) of an irredundant set on the queen's graph is difficult to characterise; known values include [4]:206–207

Pursuit–evasion game

Consider the pursuit–evasion game on an queen's graph played according to the following rules: a white queen starts in one corner and a black queen in the opposite corner. Players alternate moves, which consist of moving the queen to an adjacent vertex that can be reached without passing over (horizontally, vertically or diagonally) or landing on a vertex that is adjacent to the opposite queen. This game can be won by white with a pairing strategy.[11]

See also

References

  1. 1 2 Weisstein, Eric W. "Queen Graph". MathWorld.
  2. 1 2 3 Averbach, Bonnie; Chein, Orin (2000). Problem Solving Through Recreational Mathematics. Dover Publications. pp. 211–212. ISBN 9780486131740.
  3. Bernhardsson, Bo (1991). "Explicit Solutions to the N-Queens Problem for All N". ACM Sigart. 2 (2): 7. doi:10.1145/122319.122322. S2CID 10644706.
  4. 1 2 3 4 5 6 7 8 9 Watkins, John J. (2012). Across the Board: The Mathematics of Chessboard Problems. Princeton University Press.
  5. Dominating queens - in researchgate.net
  6. 5 Queens on a Chessboard
  7. Cockayne, E. J. (1990). "Chessboard domination problems". Discrete Mathematics. 86 (1–3): 13–20. doi:10.1016/0012-365X(90)90344-H. hdl:1828/2415.
  8. Iyer, M. R.; Menon, V. V. (1966). "On Coloring the Chessboard". The American Mathematical Monthly. 72 (7): 723.
  9. Chvátal, Václav. "Colouring the queen graphs". Retrieved 14 February 2022.
  10. Bell, Jordan; Stevens, Brett (2009). "A survey of known results and research areas for n-queens". Discrete Mathematics. 309 (1): 1–31. doi:10.1016/j.disc.2007.12.043.
  11. Averbach & Chein 2000, pp. 257–258, 443.
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