The hypothetical half-Heusler alloys VLiBi and CrLiBi containing only one transition metal element are constructed. The electronic structure and magnetic properties of VLiBi and CrLiBi are investigated by using the first-principles full-potential linearized augmented plane wave method based on density functional theory. The spin-polarized calculations of electronic structure for the half-Heusler alloys VLiBi and CrLiBi are performed. The calculation results reveal that VLiBi and CrLiBi are half-metallic ferromagnets with the half-metallic gaps of 0.25 eV and 0.46 eV and the total magnetic moments of 3.00 μB and 4.00 μB per formula unit, respectively. The total magnetic moments mainly originate from the magnetic moment on V or Cr atom. Li and Bi have small atomic magnetic moments, where the atomic magnetic moment of Bi is negative. The mean field approximation method is used to estimate the Curie temperatures of the alloys. The calculated results show that the values of Curie temperature for VLiBi and CrLiBi are 1401 K and 1551 K, respectively. To study the robustness of the half-metallicity with the change of lattice constant, the electronic structures of VLiBi and CrLiBi are also calculated under their lattice constant changing from-10% to +10% relative to the equilibrium lattice constant. It is found that the VLiBi and CrLiBi can maintain their half-metallicity and retain their total magnetic moments of 3.00 μB and 4.00 μB per formula unit even when their lattice constants change from-5.6% to 10.0% and from-6.9% to 10.0%, respectively. To discuss the effect of strongly correlated interaction on the half-metallicity, the electronic structure of VLiBi and CrLiBi are calculated by the LDA+U method with U for V-3d and Cr-3d orbital. The calculation results indicate that VLiBi and CrLiBi can keep their half-metallicity and integer total magnetic moments of 3.00 μB and 4.00 μB when the value of U reaches to 5 eV. Also, the electronic structure of VLiBi and CrLiBi are recalculated by the GGA+SOC method. The calculated results show that 1) there are some spin-down bands crossing the Fermi level, 2) the spin polarizations of VLiBi and CrLiBi at the Fermi level are 98.8% and 94.3%, respectively, and 3) total magnetic moments of VLiBi and CrLiBi are 3.03 μB and 4.04 μB per formula unit, respectively. The spin-orbit coupling has a weak effect on the half-metallic of half-Heusler alloy VLiBi and the spin polarization is still high for the half-Heusler alloy CrLiBi. The half-Heusler alloys VLiBi and CrLiBi may be useful in spintronics and other applications.