For a graph
$G$
, let
$f(G)$
denote the maximum number of edges in a bipartite subgraph of
$G$
. Given a fixed graph
$H$
and a positive integer
$m$
, let
$f(m,H)$
denote the minimum possible cardinality of
$f(G)$
, as
$G$
ranges over all graphs on
$m$
edges that contain no copy of
$H$
. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph
$H$
, there exists an
$\unicode[STIX]{x1D716}(H)>0$
such that
$f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$
. We show that, for any wheel graph
$W_{2k}$
of
$2k$
spokes, there exists
$c(k)>0$
such that
$f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$
. In particular, we confirm the conjecture asymptotically for
$W_{4}$
and give general lower bounds for
$W_{2k+1}$
.