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There are 25 students, and every student gives a chocolate to 5 other students. Is it possible for every student to receive a chocolate from exactly the 5 students they've sent chocolate to?

My first though is to construct this as a graph, with each student as a vertex so there are 25 vertices. And each vertex has a minimum degree of 5 since every student is sending out 5 chocolate.

Am I on the right track? How should I proceed next?

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    On the right track, very close to solving it. What is your next thought? – 2017-01-16
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    Let each edge represent both giving and receiving. In the graph how many odd degreed vertices are there? Is that a problem? – 2017-01-16
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    @JMoravitz Let each edge represent both giving and receiving. That means each vertex will have a degree of 5, and there are 25 vertices, so the sum of degree for this graph is 125. There is a theorem that sum of degrees of all vertices should be even, so a contradiction. – 2017-01-19
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    @user59036 exactly. – 2017-01-19

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If there are 25 students, and every student gives a chocolate to 5 other students and every student received a chocolate from exactly the 5 students they've sent chocolate to, that means that each student exchanged chocolates with 5 other students. That means, we can construct a graph, where each student is a vertex and each pair of students, that has exchanged chocolates is bound by an edge. In that case, the degree of each vertex is 5. And from this fact we can conclude, that the sum of degrees of all vertices is 125. On the other side, according to the Handshaking Lemma, if n is the number of edges of the graph, the sum of degrees of all vertices is 2n. That means, it is even. But 125 is odd. It is a contradiction. And that means, that if there are 25 students, and every student gives a chocolate to 5 other students, it is impossible for every student to receive a chocolate from exactly the 5 students they've sent chocolate to.

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    "If $n$ is the number of edges of the graph, the sum of degrees of all vertices is $2n$." I don't understand how this statement is true, could you explain more? – 2017-01-19
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    The proof of that fact you can find there: https://en.m.wikipedia.org/wiki/Handshaking_lemma – 2017-01-19