Which planar subgraph of hypercube Q4 has the maximum number of edges?

Question

I believe Q3 is the biggest planar subgraph of Q4, but I think that by drawing four edges at the outermost vertices of Q3 I can obtain a bigger subgraph. Am I thinking correctly?

Answer 1

Since Paralyzed_by_Time showed that a required subgraph has at most $28$ edges, it remains to match this upper bound by an example. For this note that $Q_4=Q_2\times Q_2$, where pairs $(v,u)$ and $(v’,u’)$ in the product are adjacent iff ($v=v’$ and $u$ is adjacent to $u’$) or ($v$ ia adjacent to $v’$ and $u=u’$). Let $Q’_2$ be the graph $Q_2$ with an edge removed. Then a graph $Q_4=Q_2\times Q’_2$ has $28$ edges and can be drawn in a plane with four nested cycles of order $4$.