Knight's graph

Key takeaways

Knight's graph belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Knight's graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Knight's graph from memory before moving on to harder problems.

Reference excerpt

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. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other. More specifically, an m × n {\displaystyle m\times n} knight's graph is a knight's graph of an m × n {\displaystyle m\times n} chessboard. Its vertices can be represented as the points of the Euclidean plane whose Cartesian coordinates ( x , y ) {\displaystyle (x,y)} are integers with 1 ≤ x ≤ m {\displaystyle 1\leq x\leq m} and 1 ≤ y ≤ n {\displaystyle 1\leq y\leq n} (the points at the centers of the chessboard squares), and with two vertices connected by an edge when their Euclidean distance is 5 {\displaystyle {\sqrt {5}}} . For an m × n {\displaystyle m\times n} knight's graph, the number of vertices is n m {\displaystyle nm} . If m > 1 {\displaystyle m>1} and n > 1 {\displaystyle n>1} then the number of edges is 4 m n − 6 ( m + n ) + 8 {\displaystyle 4mn-6(m+n)+8} (otherwise there are no edges). For an n × n {\displaystyle n\times n} knight's graph, these simplify so that the number of vertices is n 2 {\displaystyle n^{2}} and the number of edges is 4 ( n − 2 ) ( n − 1 ) {\displaystyle 4(n-2)(n-1)} . A Hamiltonian cycle on the knight's graph is a (closed) knight's tour. A chessboard with an odd number of squares has no tour, because the knight's graph is a bipartite graph (each color of squares can be used as one of two independent sets, and knight moves always change square color) and only bipartite graphs with an even number of vertices can have Hamiltonian cycles. Most chessboards with an even number of squares have a knight's tour; Schwenk's theorem provides an exact listing of which ones do and which do not. When it is modified to have toroidal boundary conditions (meaning that a knight is not blocked by the edge of the board, but instead continues onto the opposite edge) the 4 × 4 {\displaystyle 4\times 4} knight's graph is the same as the four-dimensional hypercube graph.

See also King's graph Queen's graph Rook's graph Bishop's graph Lattice graph

References

External links Weisstein, Eric W. "Knight Graph". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Knight's graph

Start with the simplest possible case. Write down what Knight's graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Knight's graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Knight's graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Knight's graph

In research: Knight's graph appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Knight's graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Knight's graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, Mathematical chess problems, Parametric families of graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Knight's graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Knight's graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Knight's graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Knight's graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Knight's graph in simple terms?

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. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other.

Why does Knight's graph matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Knight's graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Knight's graph.