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...
2026-04-23 21:05:35.1776978335
Prove that two complex are isomorphic
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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.