Structural properties of 1-planar graphs and an application to acyclic edge coloring

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph $G$ with maximum degree $\Delta(G)$ is acyclically edge $L$-choosable where $L=\max\{2\Delta(G)-2,\Delta(G)+83\}$.Comment: Please cite this published article as: X. Zhang, G. Liu, J.-L. Wu. Structural properties of 1-planar graphs and an application to acyclic edge coloring. Scientia Sinica Mathematica, 2010, 40, 1025--103

Quantum Electroweak Symmetry Breaking Through Loop Quadratic Contributions

Based on two postulations that (i) the Higgs boson has a large bare mass $m_H \gg m_h \simeq 125$ GeV at the characteristic energy scale $M_c$ which defines the standard model (SM) in the ultraviolet region, and (ii) quadratic contributions of Feynman loop diagrams in quantum field theories are physically meaningful, we show that the SM electroweak symmetry breaking is induced by the quadratic contributions from loop effects. As the quadratic running of Higgs mass parameter leads to an additive renormalization, which distinguishes from the logarithmic running with a multiplicative renormalization, the symmetry breaking occurs once the sliding energy scale $\mu$ moves from $M_c$ down to a transition scale $\mu =\Lambda_{EW}$ at which the additive renormalized Higgs mass parameter $m^2_H(M_c/\mu)$ gets to change the sign. With the input of current experimental data, this symmetry breaking energy scale is found to be $\Lambda_{EW}\simeq 760$ GeV, which provides another basic energy scale for the SM besides $M_c$. Studying such a symmetry breaking mechanism could play an important role in understanding both the hierarchy problem and naturalness problem. It also provides a possible way to explore the experimental implications of the quadratic contributions as $\Lambda_{EW}$ lies within the probing reach of the LHC and the future Great Collider.Comment: 10 pages, 2 figures, published versio

The linear arboricity of planar graphs with no short cycles

AbstractThe linear arboricity of a graph G is the minimum number of linear forests which partition the edges of G. Akiyama, Exoo and Harary conjectured that ⌈Δ(G)2⌉≤la(G)≤⌈Δ(G)+12⌉ for any simple graph G. In the paper, it is proved that if G is a planar graph with Δ≥7 and without i-cycles for some i∈{4,5}, then la(G)=⌈Δ(G)2⌉

OFDM Systems for Optical Communication with Intensity Modulation and Direct Detection

Intensity modulation and direct detection (IM/DD) is a cost-effective optical communication strategy which finds wide applications in fiber communication, free-space optical communication, and indoor visible light communication. In IM/DD, orthogonal frequency division multiplexing (OFDM), originally employed in radio frequency communication, is considered as a strong candidate solution to combat with channel distortions. In this research, we investigate various potential OFDM forms that are suitable for IM/DD channel. We will elaborate the design principles of different OFDM transmitters and investigate different types of receivers including the proposed iterative receiver. In addition, we will analyze the spectral efficiency and decoding complexities of different OFDM systems to give a whole picture of their performance. Finally, simulation results are given to assess the detection performance of different receivers