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 $$n \times m$$ king's graph is a king's graph of an $$n \times m$$ 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 $$n \times m$$ 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 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 $$n\times m$$ chessboard, there are $$(n-1)(m-1)$$ 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, $$(n-1)(m-1)-4$$ 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. 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.