The correct increasing order of lattice enthalpies for CaO, VO, and MnO in a rock-salt structure is '1'. Let's understand the reasoning behind this.

Rock-salt structure:
The rock-salt structure is a common arrangement for ionic compounds. It consists of an FCC (face-centered cubic) lattice, where cations occupy the corners of the unit cell and anions occupy the face centers. This arrangement allows for the coordination of metal ions in an octahedral manner.

Lattice enthalpy:
Lattice enthalpy is the energy required to completely separate one mole of an ionic compound into its gaseous ions. It is a measure of the ionic bond strength within a crystal lattice. The lattice enthalpy depends on factors such as the charge and size of the ions, as well as the distance between them.

Increasing order of lattice enthalpies:
To determine the increasing order of lattice enthalpies for CaO, VO, and MnO, we need to consider the factors influencing the strength of the ionic bond.

1. Charge of the ions:
The lattice enthalpy is directly proportional to the charge on the ions. Higher charge leads to stronger ionic bonds and higher lattice enthalpy. In this case, CaO has Ca2+ and O2- ions, VO has V5+ and O2- ions, and MnO has Mn2+ and O2- ions.

2. Size of the ions:
The lattice enthalpy is inversely proportional to the size of the ions. Smaller ions can approach each other more closely, resulting in stronger ionic bonds and higher lattice enthalpy. In this case, Ca2+ is larger than V5+ and Mn2+ ions.

Considering these factors, we can analyze the order of lattice enthalpies:

- MnO: Mn2+ has a higher charge than Ca2+, but the size of the Mn2+ ion is smaller than Ca2+. The higher charge is expected to increase the lattice enthalpy, but the smaller size of the ion compensates for it. Therefore, MnO has the lowest lattice enthalpy.

- VO: V5+ has the highest charge among the three ions, and its size is smaller than both Ca2+ and Mn2+ ions. The combination of a higher charge and smaller size results in a higher lattice enthalpy than MnO.

- CaO: Ca2+ has the lowest charge among the three ions, but its size is larger than both V5+ and Mn2+ ions. The lower charge is expected to reduce the lattice enthalpy, but the larger size of the ion compensates for it. Therefore, CaO has the highest lattice enthalpy.

In conclusion, the correct increasing order of lattice enthalpies for CaO, VO, and MnO in a rock-salt structure is '1', where CaO has the highest lattice enthalpy, followed by VO and MnO.