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After working for sometime I figured out the following course of action. (from a few sample cases on 4 and 5 vertices)

i) I wanted to prove that the graph had no odd degree vertex.

ii) There exists at least one vertex adjacent to all other vertices.

If I can do these, then if $n = |G|$, $(n-1)$ is even - hence, $n$ is odd.

My friend told me that by considering a typical vertex and its neighbours and considering the subgraph induced on it, he has been able to prove the 1st part.

So now to prove the 2nd part, I was cosidering a vertex with maximum degree and if it does not have the above property I wanted to derive a contradiction.

But I think I am stuck.

Any suggestions?

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You've already shown that every vertex $v$ has even degree, for if it had odd degree, than look at its set of neighbors with the induced subgraph structure, $H$. $H$ has an odd number of vertices with every vertex having odd degree, which is a contradiction.

Now, consider the adjacency matrix $A$ of $G,$ where we consider a vertex to be adjacent to itself. Then the condition of $|N(u)\cap N(v)|$ being odd translates to $A^2=F,$ where $F$ is the matrix with each entry being 1. Since every vertex of $A$ has even degree, we have the identity $AF=F$. Therefore $F^2=FA^2=F$. The identity $F^2=F$ exactly means that the number of vertices is odd. This completes the proof.

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    $\begingroup$ Very nice, but you should have added that the adjacency matrix that you construct is over F_2 rather than over Q. $\endgroup$ – darij grinberg Mar 6 '10 at 17:53
  • $\begingroup$ right you are. my bad :) $\endgroup$ – jacob Mar 6 '10 at 23:21
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    $\begingroup$ Good answer. I'd replace $FA^2$ by $A^2 F$ to make it slightly clearer. $\endgroup$ – Tony Huynh Mar 8 '10 at 19:33
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Jacob seems to have beaten me to it by a few minutes, but an algebraic graph theory proof works nicely, so I'll add my slight variant.

If $A$ is the $n \times n$ adjacency matrix of the graph, which we assume has no odd degree vertices, then over $Z_2$ the condition shows that $A^2$ has 0s on the diagonal and 1s elsewhere - i.e. that $A^2 = J - I$.

But (again over $Z_2$) $J-I$ has rank $n$ if $n$ is even. But $A$ does not have full rank as $A j = 0$ where $j$ is the all-ones vector (because graph has no odd degree vertices), and so neither can $A^2$.

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I showed this on http://www.mathlinks.ro/viewtopic.php?t=68109 . Very nice problem.

EDIT: Let me repeat the solution I gave at the above link, seeing that AoPS isn't great at printability.

I'll prove the contrapositive of the question:

Theorem 1. Let $n$ be a positive integer. Let $S$ be a graph with $2n$ vertices. Then, $S$ has two distinct vertices which have an even number of common neighbors.

Here, graphs are assumed to be finite and loopless.

Theorem 1 is problem 14.10 in Arthur Engel's Problem-Solving Strategies. My proof (more or less the same as Engel's one) relies on the following well-known fact (a particular case of the handshaking lemma):

Theorem 2. If a graph has an odd number of vertices, then it has a vertex with even degree.

Proof of Theorem 1. We assume that our graph $S$ is a simple graph (since multiple edges don't matter for this theorem). The degree of a vertex in a simple graph will mean the number of its neighbors, or, equivalently, the number of edges starting at this vertex.

We are in one of the following two cases:

Case 1: Some vertex $C$ of the graph $S$ has an odd degree.

Case 2: Every vertex of the graph $S$ has an even degree.

Let us first consider Case 1. In this case, some vertex $C$ of the graph $S$ has an odd degree. Consider such an $C$. Thus, the vertex $C$ has an odd degree, i.e., an odd number of neighbors. Let $S'$ be the subgraph of $S$ whose vertices are the neighbors of $C$ (and whose edges are only those edges of $S$ whose both endpoints are neighbors of $C$). Thus, this subgraph $S'$ has an odd number of vertices (since $S$ has an odd number of neighbors). Hence, by Theorem 2, this subgraph $S'$ must have a vertex of even degree. Consider such a vertex, and denote it by $D$. Thus, the vertex $D$ has an even degree in the subgraph $S'$. In other words, the number of neighbors of $C$ that are also neighbors of $D$ is even. In other words, the number of common neighbors of the two distinct vertices $C$ and $D$ is even. Hence, Theorem 1 is proven in Case 1.

Now, let us consider Case 2. In this case, every vertex of the graph $S$ has an even degree. Pick an arbitrary vertex $A$ of the graph $S$. (Here, we are using the fact that $n$ is a positive integer, so that $S$ has a vertex to begin with.) Construct a subgraph $S'$ of $S$ as follows:

  • The vertices of $S'$ should be all the $2n$ vertices of the graph $S$ except of the vertex $A$.

  • The edges of $S'$ should be those edges of the graph $S$ that contain a neighbor of $A$ but not the vertex $A$ itself. (In other words, an edge of $S$ is an edge of $S'$ if and only if at least one of the endpoints of this edge is a neighbor of $A$; the other endpoint can be arbitrary, but it cannot be $A$ since $A$ is not a vertex of $S'$.)

Then, the graph $S'$ has an odd number of vertices ($2n - 1$ vertices, to be precise). Hence, Theorem 2 shows that this graph $S'$ has a vertex $D$ with even degree. Consider this vertex $D$. Hence, $D$ is a vertex of $S$ distinct from $A$.

The vertex $D$ has an even degree in the graph $S$ (since we are in Case 2); in other words, there is an even number of edges of the graph $S$ that contain $D$. Let this number be $2k$ (with $k$ being a nonnegative integer). Thus, we know that exactly $2k$ edges of $S$ contain $D$.

If $D$ was a neighbor of $A$ in $S$, then all of these $2k$ edges would be edges of the subgraph $S'$, except for the edge that joins $D$ to $A$; thus, there would be a total of $2k-1$ edges in $S'$ that contain $D$; in other words, the degree of $D$ in the graph $S'$ would be $2k-1$. This would contradict the fact that the degree of $D$ in the graph $S'$ is even (since $2k-1$ is not even). Thus, $D$ cannot be a neighbor of $A$ in $S$. Therefore, the edges of $S'$ that contain $D$ are precisely the edges of $S$ that contain $D$ and a neighbor of $A$. Hence, the degree of $D$ in the subgraph $S'$ is the number of common neighbors of the vertices $A$ and $D$ in the graph $S$. Since this degree is even, we thus see that the two distinct vertices $A$ and $D$ have an even number of common neighbors. Hence, Theorem 1 is proven in Case 2.

We thus have proven Theorem 1 in both Cases 1 and 2; so the proof is complete. $\blacksquare$

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It's not true, though, is it? The graph with no vertices satisfies your property vacuously and has an even number of vertices.

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    $\begingroup$ +1, since such trivial cases may lead into trouble when taken too lightly. $\endgroup$ – darij grinberg Mar 6 '10 at 17:54
  • $\begingroup$ @darij grinberg : Yet you must show that that this is not significant : Is the empty graph an initial object in the category of graphs ? In which respect does and empty set have an even number of elements ? $\endgroup$ – Jérôme JEAN-CHARLES Nov 21 '11 at 14:36
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I proved over here that statement ii) holds when we make the stronger assumption that $|N(u) \cap N(v)|$ is exactly $1$ for every $u, v$.

Most of the argument probably does not generalize, but at least one piece of it does. That part is that if $G$ is a minimal counterexample, then the complement graph is connected. I'll prove this by contradiction:

Let $X$ and $Y$ be a partition of the vertices into two nonempty parts such that every vertex in X is connected to every vertex in Y. If $X$ has even size, then we see that $Y$ has the same property $G$ has and is smaller than $G$, so $Y$ has odd size and $G$ has an odd number of vertices. If $X$ has odd size, then if we collapse $X$ to a point $x$ we still have the property that $|N(u) \cap N(v)|$ is odd for $u, v$ in $Y$, and also for any $u$ in $Y$, $|N(x)\cap N(u)|$ is odd by your step i), so $Y\cup x$ satisfies the same properties $G$ does and contains a vertex that is connected to everything, so $Y\cup x$ has odd size, so $G$ has odd size.

Unfortunately, I can't see any obvious nice relationship that must be satisfied by two nonadjacent vertices in our graph $G$...

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