graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a king's graph is a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

that represents all legal moves of the

king King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king. *In the context of prehistory, antiquity and contemporary indigenous peoples, the ...

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

piece Piece or Pieces (not to be confused with peace) may refer to: Arts, entertainment, and media Games * Piece (chess), pieces deployed on a chessboard for playing the game of chess * ''Pieces'' (video game), a 1994 puzzle game for the Super NES * P ...

on a

chessboard A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...

where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an

n \times m

king's graph is a king's graph of an

n \times m

chessboard. It is the

map graph In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but ...

formed from the squares of a chessboard by making a vertex for each square and an edge for each two squares that share an edge or a corner. It can also be constructed as the strong product of two

path graph In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal ...

s. For an

n \times m

king's graph the total number of vertices is

n m

and the number of edges is

4nm -3(n + m) + 2

. For a square

n \times n

king's graph this simplifies so that the total number of vertices is

n^2

and the total number of edges is

(2n-2)(2n-1)

. The neighbourhood of a vertex in the king's graph corresponds to the

Moore neighborhood In cellular automata, the Moore neighborhood is defined on a two-dimensional square lattice and is composed of a central cell and the eight cells that surround it. Name The neighborhood is named after Edward F. Moore, a pioneer of cellular a ...

for cellular automata. A generalization of the king's graph, called a kinggraph, is formed from a

squaregraph In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unboun ...

(a planar graph in which each bounded face is a quadrilateral and each interior vertex has at least four neighbors) by adding the two diagonals of every quadrilateral face of the squaregraph. In the drawing of a king's graph obtained from an

n\times m

chessboard, there are

(n-1)(m-1)

crossings, but it is possible to obtain a drawing with fewer

crossings Crossings may refer to: * ''Crossings'' (Buffy novel), a 2002 original novel based on the U.S. television series ''Buffy the Vampire Slayer'' * Crossings (game), a two-player abstract strategy board game invented by Robert Abbott * ''Crossings'' ...

by connecting the two nearest neighbors of each corner square by a curve outside the chessboard instead of by a diagonal line segment. In this way,

(n-1)(m-1)-4

crossings are always possible. For the king's graph of small chessboards, other drawings lead to even fewer crossings; in particular every

2\times n

king's graph is a

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

. However, when both

n

and

m

are at least four, and they are not both equal to four,

(n-1)(m-1)-4

is the optimal number of crossings.

References

{{reflist Mathematical chess problems Parametric families of graphs

See also

References