Knight's graph
| Knight's graph | |
|---|---|
|
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 knight's tour graph is a knight's tour graph of an chessboard.[1]
For a knight's tour graph the total number of vertices is simply . For a knight's tour graph the total number of vertices is simply and the total number of edges is .[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]
References
- 1 2 Averbach, Bonnie; Chein, Orin (1980), Problem Solving Through Recreational Mathematics, Dover, p. 195, ISBN 9780486131740.
- ↑ "Sloane's A033996". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Watkins, John J. (2012), Across the Board: The Mathematics of Chessboard Problems. Paradoxes, perplexities, and mathematical conundrums for the serious head scratcher, Princeton University Press, p. 44, ISBN 9780691154985.
See also
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