King's graph

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King's graph
King's graph with white king.svg
king's graph
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Table of graphs and parameters

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

For an king's graph the total number of vertices is and the number of edges is . For a square king's graph this simplifies so that the total number of vertices is and the total number of edges is .[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]

In the drawing of a king's graph obtained from an chessboard, there are 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, crossings are always possible. For the king's graph of small chessboards, other drawings lead to even fewer crossings; in particular every king's graph is a planar graph. However, when both and are at least four, and they are not both equal to four, is the optimal number of crossings.[6][7]

See also[edit]


  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, ISBN 0-89871-513-X.
  6. ^ Klešč, Marián; Petrillová, Jana; Valo, Matúš (2013), "Minimal number of crossings in strong product of paths", Carpathian Journal of Mathematics, 29 (1): 27–32, doi:10.37193/CJM.2013.01.13, JSTOR 43999517, MR 3099062
  7. ^ Ma, Dengju (2017), "The crossing number of the strong product of two paths" (PDF), The Australasian Journal of Combinatorics, 68: 35–47, MR 3631655