Can the (infinite) graph of a tiling of the plane have no two vertices of the same degree?

I mean for each region to have three or more edges in its boundary. If there's a term for that I'd appreciate a comment as to the term. Another way to state this requirement is that the graph not have loops or parallel edges, sort of an infinite version of a simple graph.

Note that calling it a tiling is a bit of a misnomer; the regions need not be say polygonal convex. Maybe "topological tiling" or something. Each region should be bounded by a curve consisting of three or more (possibly curved) pieces joining the at least three vertices on its boundary.

I tried starting with a triangular region and making its vertices have degrees $$3,4,5$$ by its three adjoining regions, but quickly got a rather ungainly mess as I tried to make more regions while avoiding two same-degree vertices.

Any insight (or reference) appreciated. Thanks

• I fail to see why not, but an ungainly mess should be precisely what one should expect. – Paul Childs Mar 18 '19 at 7:05