We searched for a correlation between the two anomalous properties of K giants: Li enhancement and IR excess from an unbiased survey of a large sample of RGB stars. A sample of 2000 low-mass K giants with accurate astrometry from the Hipparcos catalog was chosen for which Li abundances have been determined from low-resolution spectra. Far-infrared data were collected from the $WISE$ and $IRAS$ catalogs. To probe the correlation between the two anomalies, we supplemented 15 Li-rich K giants discovered from this sample with 25 known Li-rich K giants from other studies. Dust shell evolutionary models and spectral energy distributions were constructed using the code DUSTY to estimate different dust shell properties, such as dust evolutionary time scales, dust temperatures, and mass-loss rates. Among 2000 K giants, we found about two dozen K giants with detectable far-IR excess, and surprisingly, none of them are Li-rich. Similarly, the 15 new Li-rich K giants that were identified from the same sample show no evidence of IR excess. Of the total 40 Li-rich K giants, only 7 show IR excess. Important is that K giants with Li enhancement and/or IR excess begin to appear only at the bump on the RGB. Results show that K giants with IR excess are very rare, similar to K giants with Li enhancement. This may be due to the rapid differential evolution of dust shell and Li depletion compared to RGB evolutionary time scales. We also infer from the results that during the bump evolution, giants probably undergo some internal changes, which are perhaps the cause of mass-loss and Li-enhancement events. However, the available observational results do not ascertain that these properties are correlated. That a few Li-rich giants have IR excess seems to be pure coincidence.
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