  The origin of the Li-rich K giants is still highly debated. Here, we
investigate the incidence of binarity among this family from a nine-year
radial-velocity monitoring of a sample of 11 Li-rich K giants using the HERMES
spectrograph attached to the 1.2m Mercator telescope. A sample of 13
non-Li-rich giants (8 of them being surrounded by dust according to IRAS, WISE,
and ISO data) was monitored alongside. When compared to the binary frequency in
a reference sample of 190 K giants (containing 17.4% of definite spectroscopic
binaries -- SB -- and 6.3% of possible spectroscopic binaries -- SB?), the
binary frequency appears normal among the Li-rich giants (2/11 definite
binaries plus 2 possible binaries, or 18.2% SB + 18.2% SB?), after taking
account of the small sample size through the hypergeometric probability
distribution. Therefore, there appears to be no causal relationship between Li
enrichment and binarity. Moreover, there is no correlation between Li
enrichment and the presence of circumstellar dust, and the only correlation
that could be found between Li enrichment and rapid rotation is that the most
Li-enriched K giants appear to be fast-rotating stars. However, among the dusty
K giants, the binary frequency is much higher (4/8 definite binaries plus 1
possible binary). The remaining 3 dusty K giants suffer from a radial-velocity
jitter, as is expected for the most luminous K giants, which these are.
