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Given a n-regular multigraph (multiple edges incident to the same two nodes are allowed), assume n is even. We try to assign each edge to one of its two end nodes, following a simple greedy rule: for each edge, check how many edges has been assigned to the two end nodes, and assign the edge to the end node with smaller edges. Breaking tie randomly.

Question： Assign the edges in arbitrary order, and each node will have n/2 edges assigned to them. Is it true or not? give proof or counter example.

I‘ve been stuck on this for days。 Thank you all for help！

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  • $\begingroup$ It's definitely false if n is odd. $\endgroup$
    – Tony Huynh
    Commented Jul 13, 2011 at 20:17
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    $\begingroup$ Does look like homework. $\endgroup$
    – Igor Rivin
    Commented Jul 13, 2011 at 22:31
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    $\begingroup$ Is this of interest to research mathematicians, as per the faq? I'm not convinced. Voting to close. $\endgroup$
    – Gerry Myerson
    Commented Jul 14, 2011 at 5:15

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It is false for a $C_4$ or for a cube of any dimension. And it looks like homework.

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