drawing a closed figure

mathmaniac · February 6th, 2013, 06:00 PM

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...

soroban · February 6th, 2013, 08:51 PM

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

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

mathman · February 7th, 2013, 04:08 PM

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.

Hoempa · February 8th, 2013, 12:31 AM

What is connected? All vertices >= 2? If so, do you have an example that fails [color=#105287]soroban[/color]'s conditions for traceability?

mathman · February 8th, 2013, 12:22 PM

Two triangles not connected to each other. You can draw either one, but you need to lift the pencil to do both.

mathmaniac · February 8th, 2013, 04:41 PM

Thank you soroban,but i need time to check your conjecture.

mathbalarka · February 9th, 2013, 12:48 AM

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.

mathmaniac · February 9th, 2013, 04:32 PM

Have you got any proof for it?

mathbalarka · February 9th, 2013, 08:27 PM

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).

mathmaniac · February 13th, 2013, 08:10 PM

Quote:

Originally Posted by mathbalarka

Quote:

Originally Posted by mathmaniac

Have you got any proof for it?

Of course.

Then can you post it?

February 6th, 2013, 06:00 PM	#1
mathmaniac 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...
$mathmaniac is offline$

February 8th, 2013, 12:31 AM	#4
Hoempa Math Team Joined: Apr 2010 Posts: 2,770 Thanks: 356	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?
$Hoempa is offline$

February 8th, 2013, 12:22 PM	#5
mathman Global Moderator Joined: May 2007 Posts: 6,275 Thanks: 516	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.
$mathman is offline$

February 8th, 2013, 04:41 PM	#6
mathmaniac 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.
$mathmaniac is offline$

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

drawing The closed figure without taking hands out

sketch a closed figure.

draw any closed figure

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