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.[1] 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.[2]

King's graph
${\displaystyle 8\times 8}$ king's graph
Vertices	${\displaystyle nm}$
Edges	${\displaystyle 4nm-3(n+m)+2}$
Girth	${\displaystyle 3}$ when ${\displaystyle \min(m,n)>1}$
Chromatic number	${\displaystyle 4}$ when ${\displaystyle \min(m,n)>1}$
Chromatic index	${\displaystyle 8}$ when ${\displaystyle \min(m,n)>2}$
Table of graphs and parameters

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, the total number of vertices is ${\displaystyle n^{2}}$ and the total number of edges is ${\displaystyle (2n-2)(2n-1)}$.[3]

The neighbourhood of a vertex in the king's graph corresponds to the Moore neighborhood for cellular automata.[4] 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.[5]

See also

• Knight's graph
• Rook's graph
• Lattice graph

References

1. Chang, Gerard J. (1998), "Algorithmic aspects of domination in graphs", in Du, Ding-Zhu; Pardalos, Panos M. (eds.), Handbook of combinatorial optimization, Vol. 3, Boston, MA: Kluwer Acad. Publ., pp. 339–405, MR 1665419. Chang defines the king's graph on p. 341.
2. Berend, Daniel; Korach, Ephraim; Zucker, Shira (2005), "Two-anticoloring of planar and related graphs" (PDF), 2005 International Conference on Analysis of Algorithms, Discrete Mathematics & Theoretical Computer Science Proceedings, Nancy: Association for Discrete Mathematics & Theoretical Computer Science, pp. 335–341, MR 2193130.
3. Sloane, N. J. A. (ed.). "Sequence A002943". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Smith, Alvy Ray (1971), "Two-dimensional formal languages and pattern recognition by cellular automata", 12th Annual Symposium on Switching and Automata Theory, pp. 144–152, doi:10.1109/SWAT.1971.29.
5. Chepoi, Victor; Dragan, Feodor; Vaxès, Yann (2002), "Center and diameter problems in plane triangulations and quadrangulations", Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '02), pp. 346–355, CiteSeerX 10.1.1.1.7694, ISBN 0-89871-513-X.