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

Gradient flow approach to an exponential thin film equation: global existence and latent singularity

In this work, we study a fourth order exponential equation, $u_t=\Delta e^{-\Delta u},$ derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text overlap with arXiv:1711.07405 by other author

Reevaluation of the density dependence of nucleon radius and mass in the global color symmetry model of QCD

With the global color symmetry model (GCM) at finite chemical potential, the density dependence of the bag constant, the total energy and the radius of a nucleon in nuclear matter is investigated. A relation between the nuclear matter density and the chemical potential with the action of QCD being taken into account is obtained. A maximal nuclear matter density for the existence of the bag with three quarks confined within is given. The calculated results indicate that, before the maximal density is reached, the bag constant and the total energy of a nucleon decrease, and the radius of a nucleon increases slowly, with the increasing of the nuclear matter density. As the maximal nuclear matter density is reached, the mass of the nucleon vanishes and the radius becomes infinite suddenly. It manifests that a phase transition from nucleons to quarks takes place.Comment: 18 pages, 3 figure