Two conjectures in Ramsey-Tur\'an theory

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $\phi:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Tur\'an function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,\delta n)$ for $s\in\{3,4,5\}$ and sufficiently small $\delta$, confirming a conjecture of Erd\H{o}s and S\'os from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings.Comment: 20 pages, 2 figures, 2 pages appendi

Harmonious Coloring of Trees with Large Maximum Degree

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $\Delta(T)\geq \frac{n+2}{3}$, then h(T)= \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$; \Delta(T)+1, & otherwise. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.Comment: 8 pages, 1 figur

The Role of Histone H2B Ubiquitylation and its Related Factors in Transcriptional Regulation in Mammalian Cells

Diverse histone modifications such as acetylation, methylation, and phosphorylation play important roles in transcriptional regulation throughout eukaryotes, and recent studies in yeast also have implicated H2B ubiquitylation in the transcription of specific genes. However, a systematic study of H2B ubiquitylation in mammalian cells has been hindered by the lack of information about mammalian homologues of the yeast enzymes responsible for H2B ubiquitylation. I report identification of a functional human homologue, the hBRE1A/B complex, of the yeast BRE1 E3 ubiquitin ligase. hBRE1A, which forms a complex with hBRE1B, specifically increases the global level of H2B ubiquitylation at lysine 120 and enhances activator dependent transcription in vivo. An extensive screening of cognate E2 ubiquitin conjugating enzyme for the hBRE1A/B complex revealed that hRAD6A and hRAD6B specifically interact with the N-terminal region of hBRE1A, and ubiquitylate H2B at lysine 120 in the presence of hBRE1A/B in vitro. Moreover, reduction of hBRE1A, hBRE1B and hRAD6 proteins by RNAi decreases endogenous H2B ubiquitylation, activator-dependent transcription, and both H3-K4 and H3-K79 methylation. Of special significance, I show that hBRE1A/B directly interacts with p53 and that it is recruited to the mdm2 promoter in a p53-dependent manner. These studies suggest that hBRE1A/B is an H2B-specific E3 ubiquitin ligase and that it functions, at least in part, through direct activator interactions, as a transcriptional coactivator. In addition, hBRE1A/B directly interacts with the hPAF complex to bring hRAD6 to the transcription machinery. I also found that a direct interaction between the hPAF complex and the previously characterized transcription elongation factor SII enhances their mutual association with RNA polymerase II. In an in vitro transcription assay with highly purified transcription factors, the hPAF complex and SII showed significant synergistic effects on activator- and histone acetyltransferase-dependent transcription from a chromatin template. However, addition of the H2B ubiquitylation factors to the in vitro transcription reaction led to an unexpected reduction in transcription. These results suggest that the reconstituted system lacks additional histone modifying and/or chromatin remodeling activities that link H2B ubiquitylation and gene activation. Taken together, my new findings set a stage for studying the molecular mechanisms of H2B ubiquitylation in transcriptional control in mammalian cells

Hazard area mapping during extreme rainstorms in South Korean mountains

2012 Fall.Includes bibliographical references.The concern for climate change has increased worldwide. Localized rain storms with high intensity and short duration have been observed in the United States, Europe, Australia, and China. South Korea is one of the countries that have also been impacted by extreme rainfall events during typhoons. Extreme rainstorms have caused major damage from landslides and debris flows in the South Korean mountains. The Duksan Creek watershed in South Korea was selected to simulate surface runoff using TREX during the extreme rainstorm precipitation event from July 14 to July 16, 2006. The maximum hourly rainfall was 62 mm on July 15 in 2006. The three hour rainfall from 08:00 AM to 11:00AM on this day was 168 mm. This rainstorm triggered 518 landslides and caused major infrastructure damage from debris flows. The three hour rainfall precipitation has a 100 year return period. The TREX model was calibrated in two mountainous regions of South Korea. The relative percent difference of time to peak and peak discharge on the Naerin Stream and the Naesung Stream were 6.25 %, -2.58 % and 1.90 %, -0.25 %, respectively. The TREX simulation at the Duksan Creek was performed at a 30 m resolution with distributed data on topography (DEM), soil type, and land use. The peak discharge from the TREX simulation at the Duksan Creek watershed was 452 m3/s. This value was compared to the results of several other methods and the relative percent difference was -1.1 %. The peak discharge was also compared with specific peak discharge measurements and this value corresponds to the range of values for similar watersheds. The TREX model can calculate the distribution of infiltration depth. The infiltration depth calculation typically ranged from 0.2 m to 0.3m with maximum value of 1.2 m. Based on the infinite slope analysis, such infiltration depths correspond to a critical slope angle of 25° to 29°. This range of the critical slope angle was comparable to the angle of 26° from the field investigations and from the analysis of satellite images and aerial photographs at the Duksan Creek. Several different hazard mapping methods were compared including a landslide hazard map from the Korea Forest Institute (KFRI), SINMAP, and TREX. The result of the relative predictability of TREX was slightly better an improvement of 24.6 % than the result of SINMAP. The maximum shear stress could also be calculated by the TREX model. Values of shear stress typically ranged between 0.223 kPa to 0.895 kPa in the tributaries and 1.79 kPa to 17 kPa in the main channel. Based on a critical shear stress analysis, a 1 m diameter boulder reaches incipient motion at a shear stress of 0.895 kPa