Two maxima are noticed at about $M_{\rm {bol}}\simeq-1.5$ and -4.75. They correspond to early HC-stars (HC1-HC3) and CV-stars (CV4-CV5) respectively. We recall now that the HC and CV-stars are mainly members of two distinct populations on the grounds of space distributions and kinematics, respectively the thick disk component and the old (thin) disk component (Paper II). The former sample is contaminated by a spheroidal contribution, namely the CH stars. It is thus clear that the LF of Fig. $% latex2html id marker 2873 $~\ref{lf1}$$ results from the superimposition of two LFs. This LF may be affected by the Malmquist bias as described in Knapik et al. (1998) and Paper II. It was shown there that the sample of HIPPARCOS carbon giants is reasonably complete within a sphere of radius 1.1 kpc and that the effect of the Malmquist bias is quite moderate on most photometric groups (Paper II, corrections on mean values, of a few tenths of magnitude at most, were applied in Table $% latex2html id marker 2875 $~\ref{coef_gr}$$ ). In order to minimize the effect of this bias on the Galactic LF, we selected the stars with large true parallaxes $\left(\varpi\ge0.9~\rm {mas}\right)$ and plotted the new LF with full lines in Fig. $% latex2html id marker 2879 $~\ref{lf1}.$$ It looks much like the previous one shown with dashed lines. The $M_{\rm {bol}}\simeq-4.75$ maximum is nevertheless accentuated and the minimum near $M_{\rm {bol}}\simeq-2.25$ is slightly more pronounced. The half-height width of the CV portion also drops from about 6 mag to 4.5 mag. The maximum of HC-stars appears shifted from $M_{\rm {bol}}\simeq-1.75$ to -1.25. Those modifications can be understood in terms of reducing simultaneously the effect of the Malmquist bias and the incidence of parallax errors (which are on average smaller in the $\varpi\ge0.9~\rm{mas}$ sample).

This latter LF for Galactic carbon giants in the Sun vicinity hereafter referred to as MW for Milky Way, is compared to the LFs of the Galactic bulge (MWB, 33 stars; Rich 1989), and to that of the Large Magellanic Cloud (LMC, 895 field C stars; Costa & Frogel 1996) in Fig. $% latex2html id marker 2891 $~\ref{lf2}.$$ The distance modulus 18.5 (Bergeat et al. 1998) is adopted. In Fig. $% latex2html id marker 2893 $~\ref{lf3},$$ the MW-LF is confronted to the LFs obtained for the Small Magellanic Cloud (SMC) by Azzopardi (1993), Westerlund et al. (1995), and from the data of Rebeirot et al. (1993) by Groenewegen (1998). Our MW-LF appears as larger than the LMC-LF. Errors on true parallaxes certainly contribute, while all LMC-stars are assumed at a common distance from Sun.

The most remarkable feature is the absence of LMC faint carbon stars $\left(M_{\rm {bol}}\ge -2.5\right),$ which are HC-stars in the Galactic disk. The latter thick disk population (Paper II) has no equivalent amongst the 895 field C stars of Costa & Frogel (1996), with the possible exception of two stars located in the $\left[-1.9;-1.7\right]$ and $\left[-1.3;-1.1\right]$ intervals respectively. The luminosity ranges of carbon stars in LMC clusters are still narrower (e.g. Costa & Frogel 1996). Faint LMC carbon stars were searched for with no success until now ( $M_{\rm {bol}}\le -3.3,$ Azzopardi 1999b). The discovery of 43 hot CH stars in the outer halo of the LMC (Hartwick & Cowley 1988) which are as bright as $-4\ge M_{\rm {bol}}\ge -6,$ (Feast & Whitelock 1992), is possibly part of the answer: the hot carbon stars are few in the LMC and most of them reach high luminosities. As shown in Fig. $% latex2html id marker 2905 $~\ref{lf2},$$ the carbon stars in the Galactic bulge (MWB) populate the range $1\ge M_{\rm {bol}}\ge -2.5,$ much similar to that of the early HC-stars in the thick disk ("MW'' with $M_{\rm {bol}}\ge -2.5).$ Both maxima are located at $M_{\rm {bol}}\simeq-1.25\pm0.25.$ The populations of hot carbon stars in the thick disk and Galactic bulge look similar. Very bright carbon stars $\left(M_{\rm {bol}}\simeq-6.8\right)$ were found recently in the LMC by van Loon et al. (1998, 1999). They exhibit strong mass loss and thick circumstellar shells. Absent from Fig. $% latex2html id marker 2915 $~\ref{lf2},$$ they are the counterpart of our brightest galactic CV7-objects.

Finally, the comparison to the LFs published for the SMC, is given in Fig. $% latex2html id marker 2917 $~\ref{lf3}.$$ The main maximum is located at $M_{\rm {bol}}\simeq-4.4\pm0.25,$ fainter than the $M_{\rm {bol}}\simeq-4.8\pm0.25$ value obtained for both LMC and Milky Way. A secondary maximum is marginally seen in the SMC at $M_{\rm {bol}}\simeq-2.4\pm0.4$ when averaging the three plotted LFs. It is flanked on its left by a rapid decrease. Carbon stars as faint as $M_{\rm {bol}}\simeq-1.0$ are observed. It will be interesting to check for any effect of limiting apparent magnitude. According to Azzopardi (1999b), the faint-end of the carbon star LF in SMC, as inferred from 161 stars, is $M_{\rm {bol}}\simeq-1.4,$ assuming a distance modulus of 18.8. Azzopardi et al. (1999) found in the Fornax dwarf spheroidal galaxy, low-luminosity objects as faint as $M_{\rm {bol}}\simeq-1.2,$ assuming a dust free distance modulus of 21.0. Low-metallicity systems (Z=0.004 is usually adopted for SMC) seem to have in common the existence of low-luminosity carbon stars. It is known from stellar modeling that dredge-up is favored at low metallicity (Rossi et al. 1999) when compared to the solar Z=0.02 value (e.g. Straniero et al. 1997), so that the minimum mass at which a star mixes to the envelope C-rich materials decreases with increasing Z. Most of those stars seem however too faint to have reached this advanced phase of evolution.

$\begin{figure} \par\includegraphics[width=10cm,clip]{f8new.eps} % \end{figure}$

Figure 8: The HR diagram of nearly 370 Galactic carbon and BaII giants, and related objects. Most stars populate a curved strip which correspond to theoretical evolutionary tracks of stars (RGB+AGB) with initial masses in the $0.8{-}4~M_{\odot}$ range. For Z=0.02, theoretical tracks were adapted from Bressan et al. (1993) and Marigo et al. (1996). For Z=0.008, tracks for 1 and $1.9~M_{\odot}$ labeled with asterisks were adapted from Fagotto et al. (1994c) and Marigo et al., which illustrate the resulting leftward shift. From left to right, mean values and dispersion bars are shown for the 14 carbon-rich photometric groups: HC0 (2 values: see text), HC1 HC2, HC3 and HC4 (at about the same $\log\left(T_{\rm{eff}}\right)\simeq 3.6$ ), HC5, CV1, CV2, CV3, SCV and CV4 (at about the same $\log\left(T_{\rm{eff}}\right)\simeq 3.445$ ), CV5, CV6 and CV7.

6 The luminosity function (LF) of Galactic carbon giants in the Sun vicinity