More on Rainbow Cliques in Edge-Colored Graphs

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any $\varepsilon>0$, if $e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2}$ and $k\geq 3$, then for sufficiently large $n$, the number of rainbow cliques $K_k$ in $G$ is $\Omega(n^k)$. We also characterize the extremal graphs $G$ without a rainbow clique $K_k$, for $k=4,5$, when $e(G)+c(G)$ is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).Comment: 16page

Gas kinematics and star formation in the filamentary molecular cloud G47.06+0.26

We performed a multi-wavelength study toward the filamentary cloud G47.06+0.26 to investigate the gas kinematics and star formation. We present the 12CO (J=1-0), 13CO (J=1-0) and C18O (J=1-0) observations of G47.06+0.26 obtained with the Purple Mountain Observation (PMO) 13.7 m radio telescope to investigate the detailed kinematics of the filament. The 12CO (J=1-0) and 13CO (J=1-0) emission of G47.06+0.26 appear to show a filamentary structure. The filament extends about 45 arcmin (58.1 pc) along the east-west direction. The mean width is about 6.8 pc, as traced by the 13CO (J=1-0) emission. G47.06+0.26 has a linear mass density of about 361.5 Msun/pc. The external pressure (due to neighboring bubbles and H II regions) may help preventing the filament from dispersing under the effects of turbulence. From the velocity-field map, we discern a velocity gradient perpendicular to G47.06+0.26. From the Bolocam Galactic Plane Survey (BGPS) catalog, we found nine BGPS sources in G47.06+0.26, that appear to these sources have sufficient mass to form massive stars. We obtained that the clump formation efficiency (CFE) is about 18% in the filament. Four infrared bubbles were found to be located in, and adjacent to, G47.06+0.26. Particularly, infrared bubble N98 shows a cometary structure. CO molecular gas adjacent to N98 also shows a very intense emission. H II regions associated with infrared bubbles can inject the energy to surrounding gas. We calculated the kinetic energy, ionization energy, and thermal energy of two H II regions in G47.06+0.26. From the GLIMPSE I catalog, we selected some Class I sources with an age of about 100000 yr, which are clustered along the filament. The feedback from the H II regions may cause the formation of a new generation of stars in filament G47.06+0.26.Comment: 10 pages, 11 figures, accepted for publication in A&

Pebbles in an Embedded Protostellar Disk: The Case of CB26

Planetary cores are thought to form in proto-planetary disks via the growth of dusty solid material. However, it is unclear how early this process begins. We study the physical structure and grain growth in the edge-on disk that surrounds the ~1 Myr old low-mass (~0.55 Msun) protostar embedded in the Bok Globule CB26 to examine how much grain growth has already occurred in the protostellar phase. We combine the SED between 0.9 ${\mu}$m and 6.4 cm with high angular resolution continuum maps at 1.3, 2.9, and 8.1 mm, and use the radiative transfer code RADMC-3D to conduct a detailed modelling of the dust emission from the disk and envelope of CB 26. We infer inner and outer disk radii of around 16 au and 172$\pm$22 au, respectively. The total gas mass in the disk is ~0.076 Msun, which amounts to ~14% of the mass of the central star. The inner disk contains a compact free-free emission region, which could be related to either a jet or a photoevaporation region. The thermal dust emission from the outer disk is optically thin at mm wavelengths, while the emission from the inner disk midplane is moderately optically thick. Our best-fit radiative transfer models indicate that the dust grains in the disk have already grown to pebbles with diameters of the order of 10 cm in size. Residual 8.1 mm emission suggests the presence of even larger particles in the inner disk. For the optically thin mm dust emission from the outer disk, we derive a mean opacity slope of 0.6$\pm$0.4, which is consistent with the presence of large dust grains. The presence of cm-sized bodies in the CB 26 disk indicates that solids grow rapidly already during the first million years in a protostellar disk. It is thus possible that Class II disks are already seeded with large particles and may contain even planetesimals.Comment: Accepted for publication in A&A; 17 pages, 14 figure

The chaperone activity of trigger factor is distinct from its isomerase activity during co-expression with adenylate kinase in Escherichia coli

AbstractTo investigate the molecular chaperone function of trigger factor (TF) and its relationship with isomerase activity in vivo, the assisted folding of adenylate kinase (AK) by TF in Escherichia coli was examined by measuring the amounts of soluble AK produced during co-expression. When the mutant of chicken AK, P17G, is expressed in plasmid pBVAK, 95% of the protein is found in inclusion bodies. Co-expression of AK with TF was achieved using a plasmid pBVAT that allowed expression of TF and AK in the same plasmid under separate control. Co-expression with TF resulted in an increase in the amount of soluble AK, with a higher increase when TF was expressed at higher levels in the cell. Co-expression of AK with the two TF mutants, Y221G and F233Y, in which peptidyl-prolyl cis/trans isomerase activity was 1% of wild-type, gave the same results as wild-type TF. This provides in vivo evidence that the molecular chaperone activity of TF is distinct from its isomerase activity