Primitive meteorites contain presolar grains that originated in stellar outflows and supernova ejecta. Low-density graphite grains from the Murchison carbonaceous meteorite were analyzed for the isotopic compositions of C, N, O, Mg, Si, K, Ca, and Ti by ion microprobe mass spectrometry. The grains are characterized by a large range of C-12/C-13 ratios (from 3.6 to 7200 compared to the solar ratio of 89), excesses in N-15 (N-15/N-14 up to 10 times solar) and O-18 (O-18/O-16 up to 185 times solar), large inferred Al-26/Al-27 ratios (from Mg-26 excesses) ranging up to 0.15, a large range in Si isotopic ratios (from 50% deficits in Si-29 and Si-30 relative to Si-28 up to more than 120% excesses), large excesses in K-41 and Ca-44 from the prior presence of now-extinct Ca-41 (T-1/2 = 10(5) yr) and Ti-44 (T-1/2 = 59 yr), respectively, and excesses in Ca-42, Ca-43 relative to Ca-40, and Ti-49, and Ti-50 relative to Ti-48. Several of these isotopic signatures indicate a supernova origin. In particular, the initial presence of Ti-44 and excesses of Si-28 as well as the size of the inferred Ca-41/Ca-40 ratios are proof that the carrier grains formed in supernova ejecta. We explored the possibility that the low-density graphite grains originated from C-rich ejecta of Type II supernovae. In such stars Ti-44 and Si-28 are produced in the inner layers and the presence of these two isotopes in carbonaceous grains is evidence for extensive mixing of different supernova layers in the explosion.`We performed mixing calculations of different layers of the SN models by Woosley & Weaver under the imposed boundary condition that C greater than or equal to O and compare the resulting isotopic ratios with the isotopic ratios measured in the meteoritic grains. The mixing model can explain the observed C-12/C-13, O-16/O-18, Si-30/Si-28, and Ti-44 and Ca-41 fairly well as long as jets of material from the Si-rich zone, carrying Ti-44 and pure Si-28, are assumed to penetrate the O-rich zone and are ejected into and mixed with the C-rich layers, where carbonaceous grains can form, without overwhelming these layers with the massive amounts of oxygen. Problems with the model are that it produces not enough N-15 and consistently yields lower Si-29/Si-30 ratios than those in the grains. Furthermore, large excesses in Ca-42 and Ti-50 found in several grains, which can be attributed to neutron capture, in the model are obtained only in layers with O > C. It remains to be seen whether adjustment of cross sections and/or multidimensional SN models can overcome some of these problems. It also remains to be seen whether Type Ia supernovae, which have been proposed as a source of SN grains in meteorites, can provide a better explanation. The fact that essentially all supernova grains identified so far are diamond, graphite, SiC of Type X and Si3N4, and only one oxide grain with a supernova signature has been found remains a puzzle.