  We show, for the first time, that ${\rm H_2}$ formation on dust grains can be
enhanced in disk galaxies under strong ram-pressure (RP). We numerically
investigate how the time evolution, of ${\rm H}$ {\sc i} and ${\rm H_2}$
components in disk galaxies orbiting a group/cluster of galaxies, can be
influenced by hydrodynamical interaction between the gaseous components of the
galaxies and the hot intra-cluster medium (ICM). We find that compression of
${\rm H}$ {\sc i} caused by RP increases ${\rm H_2}$ formation in disk
galaxies, before RP rapidly strips ${\rm H}$ {\sc i}, cutting off the fuel
supply and causing a drop in ${\rm H_2}$ density. We also find that the level
of this ${\rm H_2}$ formation enhancement in a disk galaxy under RP depends on
the mass of its host cluster dark matter (DM) halo, initial positions and
velocities of the disk galaxy, and disk inclination angle with respect to the
orbital plane. We demonstrate that dust growth is a key factor in the evolution
of the ${\rm H}$ {\sc i} and ${\rm H_2}$ mass in disk galaxies under strong RP.
We discuss how the correlation between ${\rm H_2}$ fractions and surface gas
densities of disk galaxies evolves with time in the galaxies under RP. We also
discuss whether or not galaxy-wide star formation rates (SFRs) in cluster disk
galaxies can be enhanced by RP if the SFRs depend on ${\rm H_2}$ densities.
