# Tilings of the plane

There are many possible tilings (or tesselations) of the plane:

What I am looking for is a general definition of what a tiling is - in terms of (topological) graph theory. That means:

Given a connected planar graph $G$ and an embedding of $G$ into the plane, i.e. a connected plane graph. What are the conditions on $G$ to be a tiling of the plane?

I won't be surprised if this definition turns out to be trivial, but I don't see it in my mind's eye, yet.

Conditions (necessary and/or sufficient) that spring to mind:

• $G$ is 2-edge-connected, i.e. every vertex/edge is contained in a cycle.

• If a (topological) connected subset of the plane contains no cycle of $G$, then it is finite.

For aesthetical reasons, I'd like to see the extra condition imposed:

• All minimal cycles of $G$ are convex.

Is there - eventually - a traceable reason for this extra condition?

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I don't understand how a single graph can be a tiling of the plane. Are you asking for the graph to be a tile, copies of which can be used to tile the plane? If so, isn't it just a question of the polygon formed by the outermost edges of the graph, what goes on inside being irrelevant? – Gerry Myerson Oct 29 '12 at 23:49
@Gerry: Thanks for your comment. I am trying to figure out how my question could be misunderstood this way. I'll try to answer your question as soon as possible. – Hans Stricker Oct 29 '12 at 23:58
So, if you take a tiling with finitely many prototiles, and take a region containing at least one of each prototile, then that region is the kind of plane graph you are looking for --- is that right? – Gerry Myerson Oct 29 '12 at 23:58
Never underestimate my ability to misunderstand a question! – Gerry Myerson Oct 29 '12 at 23:59
+1 by me: And never overestimate my ability to ask a question that cannot be misunderstood. – Hans Stricker Oct 30 '12 at 0:07