9762

Laceable Knight Graphs

This Demonstration shows how to get from any white square on a chessboard to any black square by a sequence of knight moves that visits all squares. Such a route is called a Hamiltonian path, as opposed to a Hamiltonian cycle, which starts and finishes on the same square. Drag the two locators to change the start and finish squares.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Let denote the knight graph: knight moves on an board, where is always assumed. The graph is always bipartite. A bipartite graph having the property that one can get from any point in one part to any point in the other part using a Hamiltonian path is called Hamilton-laceable. This Demonstration shows that is Hamilton-laceable, thus extending the very old result that is Hamiltonian.
By a theorem of A. Schwenk [1], the collection of graphs that admit Hamiltonian cycles consists of with and even; with odd, , and even; and with even and . Of these, I have shown that the following are not laceable: , , , , and , while the following are laceable: , , , , , and . Thus, one conjecture is that the Hamiltonian graph is laceable iff .
Reference
[1] A. Schwenk, "Which Rectangular Chessboards Have a Knight’s Tour?," Mathematics
Magazine,
64(5), 1991 pp. 325–332. www.jstor.org/stable/2690649.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+