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

List (d,1)-total labelling of graphs embedded in surfaces

The (d,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we consider the list version of (d,1)-total labelling of graphs. Let G be a graph embedded in a surface with Euler characteristic $\epsilon$ whose maximum degree $\Delta(G)$ is sufficiently large. We prove that the (d,1)-total choosability $C_{d,1}^T(G)$ of $G$ is at most $\Delta(G)+2d$.Comment: 6 page

Characterizations of maximum fractional (g,f)-factors of graphs

AbstractIn this paper a characterization of maximum fractional (g,f)-factors of a graph is presented. The properties of the maximum fractional (g,f)-factors and fractional (g,f)-factors with the minimum of edges are also given, generalizing the results given in [William Y.C. Chen, Maximum (g,f)-factors of a general graph, Discrete Math. 91 (1991) 1–7] and [Edward R. Scheinerman, Daniel H. Ullman, Fractional Graph Theory, John Wiley and Sonc, Inc., New York, 1997]. Furthermore, some new results on fractional factors are obtained which may be used in the design of networks. A polynomial time algorithm can be obtained for actually finding such maximum fractional (g,f)-factors in a graph from the proof

Edge covering pseudo-outerplanar graphs with forests

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or two forests and a matching, or $\max\{\Delta(G),4\}$ matchings, or $\max\{\lceil\Delta(G)/2\rceil,3\}$ linear forests. These results generalize some ones on outerplanar graphs and $K_{2,3}$-minor-free graphs, since the class of pseudo-outerplanar graphs is a larger class than the one of $K_{2,3}$-minor-free graphs.Comment: This paper was done in the winter of 2009 and has already been submitted to Discrete Mathematics for 3rd round of peer revie

Performance of an Inertially Coupled, 3-Mode Gravitational-Wave Antenna Prototype

A prototype three‐mode gravitational wave antenna which employs a two‐mode torsional transducer has been constructed and tested. For the torsional transducer the coupling from one stage to the next is via inertial forces, whereas in a conventional transducer the stage‐to‐stage coupling is proportional to the relative displacements via the springs. Experiments with our antenna‐torsional transducer prototype demonstrate a maximum antenna bandwidth of 260 Hz (29% of the antenna resonant frequency of 900 Hz) and a mechanical amplification factor of 40. A mathematical model for the three‐mode antenna has been developed and predictions of the system transfer functions and transient response are in close agreement with the measurements. Through the optimization of the transducer parameters we find that maximum fractional antenna bandwidths near 30% may be simultaneously achieved with mechanical amplification factors of 100 or more. Furthermore, the torsional transducer has a larger mechanical gain‐antenna bandwidth product than a linear transducer with similar masses

Visual Analytics Law Enforcement Toolkit

VALET, visual analytics law enforcement toolkit, is an interactive toolkit developed for law enforcement agencies to explore concerned crime information and make police resource allocation strategies. As a visual analytics toolkit, VALET is coupled with data collection, data analytics and data prediction. The objective of VALET is to assist law enforcement agencies to reduce crime rate by wisely allocating police resource based on the analytics of historical crime records. The program incorporates three steps to generate police patrol route and policeman allocation. The first step is to generate crime hotspots and crime contours of collected crime data. The next step is to analyze historical crime information and predict potential defects. Finally, the program is to compute police patrol routes and allocate police resource based on schedule and specialty. The results from the program allow us to generate risky area for different type of crimes, and evaluate policemen’s performance in dealing with different type of crimes. Thus, police department is able to assign police officers to designed patrol routes that suggested by prediction tool based on policemen’s specialty. This would take advantage of crime prediction and decrease the time of handling criminal activities. With VALET, law enforcement agencies are able to explore concerned crime information intelligently. At the same time, police department is prompted to allocate police resource wisely