Knight's graph

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Knight's graph
Knight's graph.svg
8x8 Knight's graph
Vertices nm
Edges 4mn-6(m+n)+8
Girth 4 (if n≥3, m≥ 5)

In graph theory, a knight's graph, or a knight's tour graph, is a graph that represents all legal moves of the knight 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 knight's tour graph is a knight's tour graph of an n \times m chessboard.[1]

For a n \times m knight's tour graph the total number of vertices is simply nm. For a n \times n knight's tour graph the total number of vertices is simply n^2 and the total number of edges is 4(n-2)(n-1).[2]

A Hamiltonian path on the knight's tour graph is a knight's tour.[1] Schwenk's theorem characterizes the sizes of chessboard for which a knight's tour exist.[3]

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