  We show that any $3$-connected cubic plane graph on $n$ vertices, with all
faces of size at most $6$, can be made bipartite by deleting no more than
$\sqrt{(p+3t)n/5}$ edges, where $p$ and $t$ are the numbers of pentagonal and
triangular faces, respectively. In particular, any such graph can be made
bipartite by deleting at most $\sqrt{12n/5}$ edges. This bound is tight, and we
characterise the extremal graphs. We deduce tight lower bounds on the size of a
maximum cut and a maximum independent set for this class of graphs. This
extends and sharpens the results of Faria, Klein and Stehlik [SIAM J. Discrete
Math. 26 (2012) 1458-1469].
