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 February 6th, 2013, 06:00 PM #1 Senior Member   Joined: Dec 2012 Posts: 450 Thanks: 0 drawing a closed figure What ensures if you can draw a closed figure without lifting your hand off and without repeating through a line? I tried different conjectures but all of them failed for some figures. Please help me out...
February 6th, 2013, 08:51 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 407

Re: drawing a closed figure

Hello, mathmaniac!

Quote:
 What ensures if you can draw a closed figure without lifting your hand off and without repeating through a line?

An even vertex has an even number of paths leading out of it.
An odd vertex has an odd number of paths leading out of it.

If all the vertices are even, the figure is traceable.

If there are any odd vertices, there must be exactly two of them.
Otherwise, the figure is not traceable.

Code:
              B
A o * * * o * * * o C
*     *   *     *
*   *       *   *
* *           * *
H o               o D
* *           * *
*   *       *   *
*     *   *     *
G o * * * o * * * o E
F
$A,C,E,G\text{ each have 2 paths.}
B,D,F,H\text{ each have 4 paths.}
\text{All vertices are even.}
\text{The figure is traceable.}$

Code:
            A
o
*   *
*       *
E o * * * * * o B
* *       * *
*   *   *   *
*     o     *
*   * F *   *
* *       * *
D o * * * * * o C
$A,B,E,F\text{ are even.}
C,D\text{ are odd.}
\text{The figure is traceable.}$

Code:
    A o * * * * * o B
*           *
*     H     *
G o * * o * * o C
*     *     *
*     *     *
F o * * o * * o D
E
$A,B,D,F\text{ are even.}
C,E,G,H\text{ are odd.}
\text{The figure is }not\text{ traceable.}$

February 7th, 2013, 04:08 PM   #3
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Posts: 6,497
Thanks: 580

Re: drawing a closed figure

Quote:
 An even vertex has an even number of paths leading out of it. An odd vertex has an odd number of paths leading out of it. If all the vertices are even, the figure is traceable. If there are any odd vertices, there must be exactly two of them. Otherwise, the figure is not traceable.
You need also that the figure is connected.

 February 8th, 2013, 12:31 AM #4 Math Team   Joined: Apr 2010 Posts: 2,778 Thanks: 361 Re: drawing a closed figure What is connected? All vertices >= 2? If so, do you have an example that fails [color=#105287]soroban[/color]'s conditions for traceability?
 February 8th, 2013, 12:22 PM #5 Global Moderator   Joined: May 2007 Posts: 6,497 Thanks: 580 Re: drawing a closed figure Two triangles not connected to each other. You can draw either one, but you need to lift the pencil to do both.
 February 8th, 2013, 04:41 PM #6 Senior Member   Joined: Dec 2012 Posts: 450 Thanks: 0 Re: drawing a closed figure Thank you soroban,but i need time to check your conjecture.
February 9th, 2013, 12:48 AM   #7
Math Team

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From: India, West Bengal

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Thanks: 86

Math Focus: Number Theory
Re: drawing a closed figure

Quote:
 Originally Posted by mathmaniac Thank you soroban,but i need time to check your conjecture.
It's no conjecture. It's the basics of graph theory.

 February 9th, 2013, 04:32 PM #8 Senior Member   Joined: Dec 2012 Posts: 450 Thanks: 0 Re: drawing a closed figure Have you got any proof for it?
February 9th, 2013, 08:27 PM   #9
Math Team

Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: drawing a closed figure

Quote:
 Originally Posted by mathmaniac Have you got any proof for it?
Of course.

Quote:
 Originally Posted by soroban If there are any odd vertices, there must be exactly two of them.
Um, soroban, you are talking about Eulerian graphs, right? But not every traceable graph is Hamiltonian.

Shouldn't we have even number of odd vertices for Hamiltonian ones? i.e. GP(5,2) is the Peterson graph. Where the number of odd nodes are 10 (not 2).

February 13th, 2013, 08:10 PM   #10
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Joined: Dec 2012

Posts: 450
Thanks: 0

Re: drawing a closed figure

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by mathmaniac Have you got any proof for it?
Of course.
Then can you post it?

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