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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:
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 Code: A o * * * * E o * * * * * o B * * * * * * * * * o * * * F * * * * * * D o * * * * * o C Code: A o * * * * * o B * * * H * G o * * o * * o C * * * * * * F o * * o * * o D E  
February 7th, 2013, 04:08 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,344 Thanks: 534  Re: drawing a closed figure Quote:
 
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: 6,344 Thanks: 534  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 Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: drawing a closed figure Quote:
 
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:
Quote:
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  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: drawing a closed figure Quote:
 

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