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321 Galactic fundamental-mode Cepheids with good B,V,I photometry byBerdnikov et al. 2000 and with homogenized color excesses E(B-V) based onFernie et al. 1995 are used to determine their P-C relation in the range0.4<log P<1.6. Distances of 25 Cepheids in open clusters (Feast 1999) and of 28Cepheids with BBW distances (Gieren et al. 1998) are used in a first step todetermine the absorption coefficients R_B=4.17, R_V=3.17, and R_I=1.89. The twosets define two independent P-L relations which agree very well in slope. Theyare therefore combined into a single mean Galactic P-L relation. GalacticCepheids are redder in (B-V) than those in LMC and SMC as shown by the over1000 Cloud Cepheids with good standard B,V,I photometry by Udalski et al 1999.Also the (B-V), (V-I) two-color diagrams differ between Cepheids in the Galaxyand the Clouds, attributed both to the effects of metallicity differences, onthe spectral energy distributions of the Cepheids and to a shift in theeffective temperature of the middle of the instability strip for LMC and SMCrelative to the Galaxy by about Delta log Te ~ 0.02 at M_V=-4.0 mag, hotter forboth LMC and SMC. Differences in the P-C relations between the Galaxy and theClouds show that there cannot be a universal P-L relation from galaxy-to-galaxyin any given color. The inferred non-uniqueness of the slope of the P-Lrelations in the Galaxy, LMC, and SMC is born out by the observations. TheCloud Cepheids follow a shallower overall slope of the P-L relations in B,V,Ithan the Galactic ones. LMC and SMC Cepheids are brighter in V than in theGalaxy by up to 0.5 mag at short periods and fainter at long periods.Comment: 29 pages, 19 figures, 7 tables. Accepted for publication in Astronomy  and Astrophysic

New Period-Luminosity and Period-Color relations of classical Cepheids: I. Cepheids in the Galaxy