King's graph

In graph theory, a king's graph is a graph that represents all legal moves of the king chess piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an ${\displaystyle n\times m}$king's graph is a king's graph of an ${\displaystyle n\times m}$chessboard. It is the map graph 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 graphs.

For an ${\displaystyle n\times m}$king's graph the total number of vertices is ${\displaystyle nm}$and the number of edges is ${\displaystyle 4nm-3(n+m)+2}$. For a square ${\displaystyle n\times n}$king's graph this simplifies so that the total number of vertices is ${\displaystyle n^{2}}$and the total number of edges is ${\displaystyle (2n-2)(2n-1)}$.

The neighbourhood of a vertex in the king's graph corresponds to the Moore neighborhood for cellular automata. A generalization of the king's graph, called a kinggraph, is formed from a squaregraph (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 ${\displaystyle n\times m}$chessboard, there are ${\displaystyle (n-1)(m-1)}$crossings, but it is possible to obtain a drawing with fewer 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, ${\displaystyle (n-1)(m-1)-4}$crossings are always possible. For the king's graph of small chessboards, other drawings lead to even fewer crossings; in particular every ${\displaystyle 2\times n}$king's graph is a planar graph. However, when both ${\displaystyle n}$and ${\displaystyle m}$are at least four, and they are not both equal to four, ${\displaystyle (n-1)(m-1)-4}$is the optimal number of crossings.

• Knight's graph
• Queen's graph
• Rook's graph
• Bishop's graph
• Lattice graph
• Chess portal