# 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
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]

King's graph | |
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king's graph | |

Vertices | |

Edges | |

Girth | when |

Chromatic number | when |

Chromatic index | when |

Table of graphs and parameters |

For a king's graph the total number of vertices is and the number of edges is . For a square king's graph, 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]

## See also

## References

- 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. - 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. - Sloane, N. J. A. (ed.). "Sequence A002943".
*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. - 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. - 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, ISBN 0-89871-513-X.