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Thread: Graph Theory

  1. #1
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    Graph Theory

    Let G be a r-regular graph on n vertices with e edges. Prove one of n or r must divide e.

    The only thing that I can think of to apply to this problem is that a r-regular graph has every vertex at r degrees.



    Thanks in advance!!
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  2. #2
    Junior Member guildmage's Avatar
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    Maybe you can use the handshaking lemma for this. Recall that $\displaystyle \sum\limits_{v \in V} {d(v)} = 2e$, where $\displaystyle d(v)$ is the degree of a vertex $\displaystyle v$. Since the graph is r-regular and has n vertices, then we have $\displaystyle nr=2e$. But if $\displaystyle n$ and $\displaystyle r$ do not divide $\displaystyle e$ (at the same time). Then $\displaystyle nr$ does not divide $\displaystyle e$. This is a contradiction. Thus either $\displaystyle n$ or $\displaystyle r$ must divide $\displaystyle e$.
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