Hi guys, I'm new here.
Well, actually I'm studying graph theory and the follow question is driving me crazy. Any hint in any direction would be appreciated.


Here is the question:


Let $G = G[X, Y]$ a bipartite graph in which each vertex in X is of odd degree. Suppose at any two vertices of X have an even number of common neighbours.
Show that G has matching covering every vertex of X.



Matching in bipartite graphs
Is it a homework problem? (If so, it is a nice one and new to me.) So you need to rule out the existence of a set $A \subseteq X$ such that the set $B$ of all $y \in Y$ adjacent to some $a \in A$ has strictly smaller size. Let $|B|=m$ and think of the neighbors of each $a \in A$ as a vector in $\mathbb{Z}\_2^m.$ Each pair of these vecotrs is orthogonal in that their dot product is zero in $\mathbb{Z}\_2.$ See what that tells you.


