queen's graph

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 $1\backslash times\; n$ and $2\backslash times\; 2$ queen's graphs are complete.{{mathworld|title=Queen Graph|urlname=QueenGraph}}

{{Chess diagram small

| tright

|

|__|__|ql|__|__|__|__|__

|__|__|__|__|ql|__|__|__

|__|ql|__|__|__|__|__|__

|__|__|__|__|__|__|__|ql

|ql|__|__|__|__|__|__|__

|__|__|__|__|__|__|ql|__

|__|__|__|ql|__|__|__|__

|__|__|__|__|__|ql|__|__

| An independent set of size 8 for an $8\backslash times\; 8$ chessboard (such sets are necessarily also dominating).

}}

{{Main|Eight queens puzzle}}

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 $n\backslash times\; n$ chessboard, the largest independent set contains at most n vertices, as no two queens can be in the same row or column.{{cite book|title=Problem Solving Through Recreational Mathematics|title-link=Problem Solving Through Recreational Mathematics|last1=Averbach|first1=Bonnie|author1-link=Bonnie Averbach|last2=Chein|first2=Orin|publisher=Dover Publications|year=2000|isbn=9780486131740|pages=211–212}} This upper bound can be achieved for all n except n=2 and n=3.{{cite journal |last=Bernhardsson |first=Bo |date=1991 |title=Explicit Solutions to the N-Queens Problem for All N |journal=ACM SIGART Bulletin |volume=2 |issue=2 |pages=7 |doi=10.1145/122319.122322 |doi-access=free |s2cid=10644706}} In the case of n=8, this is the traditional eight queens puzzle.

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 $8\backslash times\; 8$ chessboard, five queens can dominate, and this is the minimum number possible{{rp|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{{rp|113–114}} (e.g. from a1 to h8), or all on a subdiagonal (e.g. from a2 to g8).[https://www.researchgate.net/figure/Dominating-sets-of-queens-on-a-standard-chessboard_fig3_35457519 Dominating queens - in researchgate.net][https://www.pleacher.com/mp/puzzles/miscpuz3/mobquen5.html 5 Queens on a Chessboard]

Modifying the graph by replacing the non-looping rectangular $8\backslash times\; 8$ chessboard with a torus or cylinder reduces the minimum dominating set size to four.{{cite book|title=Across the Board: The Mathematics of Chessboard Problems|last=Watkins|first=John J.|year=2012|publisher=Princeton University Press}}{{rp|139}}

{{Chess diagram 5x5

| tright

|

|xo|__|xo|__|xo

|__|xo|xo|xo|__

|xo|xo|xx|xo|xo

|__|xo|xo|xo|__

|xo|__|xo|__|xo

| Dotted squares are adjacent to the centre square. The 8 non-adjacent squares are adjacent in the corresponding knight's graph.{{rp|117}}

}}

The $3\backslash times\; 3$ queen's graph is dominated by the single vertex at the centre of the board. The centre vertex of the $5\backslash times\; 5$ queen's graph is adjacent to all but 8 vertices: those vertices that are adjacent to the centre vertex of the $5\backslash times\; 5$ knight's graph.{{rp|117}}

Define the domination number d(n) of an $n\backslash times\; n$ 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 $d(n)\backslash le\; dd(n)$ for all n. The bound is attained for $d(8)=dd(8)=5$, but not for $d(11)=5,dd(11)=7$.{{rp|119}}

The domination number is linear in n, with bounds given by:{{rp|119, 121}}

: $\backslash frac\{n-1\}\{2\}\; \backslash le\; d(n)\backslash le\; n-\backslash left\; \backslash lfloor\; \backslash frac\{n\}\{3\}\backslash right\; \backslash rfloor.$

Initial values of d(n), for $n=1,2,3,\backslash dots$, are 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5 {{OEIS|id=A075458}}.

Let K_n be the maximum size of a subset of $\backslash \{1,2,3,\backslash dots,n\backslash \}$ 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 $n\backslash times\; n$ queen's graph is $n-K\_n$.{{rp|116}}

Define the independent domination number ID(n) to be the size of the smallest independent, dominant set in an $n\backslash times\; n$ queen's graph. It is known that .{{cite journal|title=Chessboard domination problems|journal=Discrete Mathematics|last=Cockayne|first=E. J.|volume=86|issue=1–3|year=1990|pages=13–20|doi=10.1016/0012-365X(90)90344-H|doi-access=free|hdl=1828/2415|hdl-access=free}}

File:Queen's graph colouring.jpg of the $8\backslash times\; 8$ queen's graph.{{cite journal|title=On Coloring the $n\backslash times\; n$ Chessboard|journal=The American Mathematical Monthly|last1=Iyer|first1=M. R.|last2=Menon|first2=V. V.|volume=72|issue=7|year=1966|page=723}} 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 $n\backslash times\; n$ 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). The chromatic number is exactly n if $n\backslash equiv\; 1,5\; \backslash pmod\; 6$ (i.e. n is one more or one less than a multiple of 6).{{cite web|url=http://users.encs.concordia.ca/~chvatal/queengraphs.html|title=Colouring the queen graphs|last=Chvátal|first=Václav|author-link=Václav Chvátal|accessdate=14 February 2022}}

The chromatic number of an $8\backslash times\; 8$ queen's graph is 9.{{cite journal|title=A survey of known results and research areas for n-queens|journal=Discrete Mathematics|last1=Bell|first1=Jordan|last2=Stevens|first2=Brett|volume=309|issue=1|year=2009|pages=1–31|doi=10.1016/j.disc.2007.12.043|doi-access=free}}

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 $n\backslash times\; n$ queen's graph is difficult to characterise; known values include $IR(5)=5,IR(6)=7,IR(7)=9,IR(8)=11.${{rp|206–207}}

Consider the pursuit–evasion game on an $8\backslash times\; 8$ 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.{{sfn|Averbach|Chein|2000|pages=257–258, 443}}

• King's graph
• Knight's graph
• Rook's graph
• Bishop's graph
• {{portal-inline|Chess}}

{{reflist}}

Category:Mathematical chess problems

Category:Parametric families of graphs