In the article Tilings (by Ardila and Stanley) I read that a necessary and sufficient condition for a domino tiling to exist for some region with a checkerboard coloring, is that each group of $k$ white cells has at least $n \geq k$ black neighbors among them in the region.
The article then mentions that this is a specific case of the more general marriage theorem. I wanted to see if I can find a proof of the specific version, without introducing external concepts (such as representatives or neighborhoods); just using concepts that are already part of my tiling setup.**
I managed to specialize a proof, but it was very clunky until I introduced the following two concepts: a white patch is a set of white cells and all their neighbors. A white patch $P$ is bad if $|B(P)| < |W(P)|$ (where $W(P)$ is the set of white cells and $B(P)$ the set of black cells in $P$.) Similarly we can define a black patch and a bad black patch.
These concepts turn out to be useful not only in proving the theorem, but also in applications. For example, to prove that a geometric operation on a region does not affect its tileability we only need to show that the operation preserves bad patches. (I mean with "useful", easier to talk about; it's not like it's a new idea or to leads to new results AFAICS).
What I want to know is: Is there standard terminology for patches and bad patches, perhaps from graph theory or combinatorics? I mean, are these concepts already in use under different names?
** This is similar in spirit to what I was doing in Elementary proof of transformations of domino tilings.
