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We report an enhancement of thermal conductivity ($\kappa$) below thesuperconducting transition temperature ($T_C$) in the high carrier density$\beta$-V$_{1-x}$Ti$_x$ alloys. We find that the point defects generated whenTi is added to V reduce the electron mean free path down to the inter-atomicdistances and make the high frequency phonons ineffective in carrying heat. Inthis Mott-Ioffe-Regel limit, the phonon thermal conductivity is dominated bythe low frequency phonons limited by the scattering due to the electrons. Theformation of Cooper pairs below the $T_C$ re-normalizes the phonon mean freepath and enhances the $\kappa$.

Thermal conductivity of strong coupling V$_{1-x}$Ti$_x$ superconductors in the Mott-Ioffe-Regel limit