Prove that two complex are isomorphic Let $A:=$ matching complex of bipartite graph ($m \times n$ size)
$B := m \times n$ chess board complex 
Show that $A$ and $B$ are isomorphic.
I don't know exact definition of isomorphic... so i can't solve... 

An isomorphism is a bijection that preserves adjacencies. So you want a bijection $f: A \to B$. Here is a hint- a chess-board is two-colored, right? No adjacent squares have the same color. I would look at a bijection this way. How many white squares on the chess board do you have? If you have $m$ white squares, then the vertex partition with $m$ vertices in $A$ is isomorphic to the set of white squares. 
