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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3$\begingroup$ 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. $\endgroup$– Timothy BuddCommented Aug 28, 2020 at 6:41
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2 Answers
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To supplement Gordon's answer based on my comment, here's a planar drawing:
answered Aug 28, 2020 at 11:17
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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
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]
answered Aug 28, 2020 at 8:13
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$\begingroup$ @GerryMyerson It is planar, else Sage would complain if I tried to get its faces, Timothy Budd has provided the sensible plane drawing... $\endgroup$ Commented Aug 28, 2020 at 11:45
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