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Is there a plane graph such that

(1) the outer face has degree 3, i.e, is a triangle,

(2) every inner face has degree 5, and

(3) any two degree 5 faces share at most one commong edge.

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    $\begingroup$ If you have a plane graph with a pair of inner faces of degree 5 that share more than one edge, you can simply "insert" a dodecahedron into one (or both) of the faces to alleviate the situation. So an example should be easy to construct. $\endgroup$ Aug 28, 2020 at 6:41

2 Answers 2

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To supplement Gordon's answer based on my comment, here's a planar drawing:

Pentagulation of the triangle

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Using Timothy Budd's suggestion, it is easy to find such a graph by gluing two dodecahedra together. Here is the SageMath code to make this graph.

d1 = graphs.DodecahedralGraph()
d2 = graphs.DodecahedralGraph()
h = d1.disjoint_union(d2)
h.merge_vertices([(0,0),(1,0)])
h.merge_vertices([(0,1),(1,1)])
h.merge_vertices([(0,2),(1,2)])
h.add_edge((0,3),(1,3))
h.relabel()

Here is the picture

enter image description here

If you need to check that everything is satisfied then the below code finds all the 5-faces as sets of vertex-pairs (hence the nested "Set" stuff) and then checks all the pairwise intersections.

f5s = [Set([Set(e) for e in f]) for f in h.faces() if len(f) == 5]
[len(x.intersection(y)) for x in f5s for y in f5s if x != y]
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    $\begingroup$ This is a plane graph? $\endgroup$ Aug 28, 2020 at 10:59
  • $\begingroup$ @GerryMyerson It is planar, else Sage would complain if I tried to get its faces, Timothy Budd has provided the sensible plane drawing... $\endgroup$ Aug 28, 2020 at 11:45
  • $\begingroup$ OK, very good.. $\endgroup$ Aug 28, 2020 at 12:27

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