L(2, 1)-Labeling Halin Graphs with Maximum Degree Eight

Suppose that T is a plane tree without vertices of degree 2 and with at least one vertex of at least degree 3, and C is the cycle obtained by connecting the leaves of T in a cyclic order. Set G=T∪C, which is called a Halin graph. A k-L(2,1)-labeling of a graph G=(V,E) is a mapping f:V(G)→{0,1,…,k} such that, for any x1,x2∈V(G), it holds that |f(x1)−f(x2)|≥2 if x1x2∈E(G), and |f(x1)−f(x2)|≥1 if the distance between x1 and x2 is 2 in G. The L(2,1)-labeling number, denoted λ(G), of G is the least k for which G is k-L(2,1)-labelable. In this paper, we prove that every Halin graph G with Δ=8 has λ(G)≤10. This improves a known result, which states that every Halin graph G with Δ≥9 satisfies λ(G)≤Δ+2. This result, together with some known results, shows that every Halin graph G satisfies λ(G)≤Δ+6

<i>L</i>(2, 1)-Labeling Halin Graphs with Maximum Degree Eight

Suppose that T is a plane tree without vertices of degree 2 and with at least one vertex of at least degree 3, and C is the cycle obtained by connecting the leaves of T in a cyclic order. Set G=T∪C, which is called a Halin graph. A k-L(2,1)-labeling of a graph G=(V,E) is a mapping f:V(G)→{0,1,…,k} such that, for any x1,x2∈V(G), it holds that |f(x1)−f(x2)|≥2 if x1x2∈E(G), and |f(x1)−f(x2)|≥1 if the distance between x1 and x2 is 2 in G. The L(2,1)-labeling number, denoted λ(G), of G is the least k for which G is k-L(2,1)-labelable. In this paper, we prove that every Halin graph G with Δ=8 has λ(G)≤10. This improves a known result, which states that every Halin graph G with Δ≥9 satisfies λ(G)≤Δ+2. This result, together with some known results, shows that every Halin graph G satisfies λ(G)≤Δ+6

Diversity and functional prediction of microbial communities involved in the first aerobic bioreactor of coking wastewater treatment system.

The pre-aerobic process of coking wastewater treatment has strong capacity of decarbonization and detoxification, which contribute to the subsequent dinitrogen of non-carbon source/heterotrophic denitrification. The COD removal rate can reach > 90% in the first aerobic bioreactor of the novel O/H/O coking wastewater treatment system during long-term operation. The physico-chemical characteristics of influent and effluent coking wastewater in the first aerobic bioreactor were analyzed to examine how they correlated with bacterial communities. The diversity of the activated sludge microbial community was investigated using a culture-independent molecular approach. The microbial community functional profiling and detailed pathways were predicted from the 16S rRNA gene-sequencing data by the PICRUSt software and the KEGG database. High-throughput MiSeq sequencing results revealed a distinct microbial composition in the activated sludge of the first aerobic bioreactor of the O/H/O system. Proteobacteria, Bacteroidetes, and Chlorobi were the decarbonization and detoxification dominant phyla with the relative abundance of 84.07 ± 5.45, 10.89 ± 6.31, and 2.96 ± 1.12%, respectively. Thiobacillus, Rhodoplanes, Lysobacter, and Leucobacter were the potential major genera involved in the crucial functional pathways related to the degradation of phenols, cyanide, benzoate, and naphthalene. These results indicated that the comprehensive understanding of the structure and function diversity of the microbial community in the bioreactor will be conducive to the optimal coking wastewater treatment

Screening genetically modified organisms using multiplex-PCR coupled with oligonucleotide microarray

Abstract In this research, we developed a multiplex polymerase chain reaction (multiplex-PCR) coupled with a DNA microarray system simultaneously aiming at many targets in a consecutive reaction to detect a genetically modified organism (GMO). There are a total of 20 probes for detecting a GMO in a DNA microarray which can be classified into three categories according to their purpose: the first for screening GMO from un-transgenic plants based on the common elements such as promoter, reporter and terminator genes; the second for specific gene confirmation based on the target gene sequences such as herbicide-resistance or insect-resistance genes; the third for species-specific genes which the sequences are unique for different plant species. To ensure the reliability of this method, different kinds of positive and negative controls were used in DNA microarray. Commercial GM soybean, maize, rapeseed and cotton were identified by means of this method and further confirmed by PCR analysis and sequencing. The results indicate that this method discriminates between the GMOs very quickly and in a cost-saving and more time efficient way. It can detect more than 95% of currently commercial GMO plants and the limits of detection are 0.5% for soybean and 1% for maize. This method is proved to be a new method for routine analysis of GMOs