High-resolution spectra were used to derive CNO abundances and the C-1
2/C-13 ratios for the very Li-rich K giants HD 19745 and HD 39853. O a
nd C-12/C-13 are also obtained for comparison for the Li-rich K giant
HD 787. Results from the literature for other very Li-rich giants were
also gathered to allow for a more complete discussion. The three most
Li-rich giants known, HD 19745, HD 39853, and HD 95799, with epsilon(
Li) greater than or equal to 3.0, show low C-12/C-13 values (C-12/C-13
less than or equal to 15), requiring an extra-mixing mechanism, where
as HD 9746 and HD 112127 with epsilon(Li) similar to 2.7 show C-12/C-1
3 similar to 25, following the standard first dredge-up predictions. T
he data for these five stars indicate that no dear correlation is foun
d between C-12/C-13 and epsilon(Li) for the very Li-rich giants. An im
portant conclusion is that, since for the three most Li-rich giants, C
-12/C-13 is low, efficient mixing should have lowered both C-12/C-13 a
nd Li abundances. The very high Li abundances actually found in these
stars call therefore for an independent mechanism enriching the Li.