An edge-colored graph is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph , denoted by , is the smallest number of colors that are needed to color the edges of in order to make it proper connected. In this paper, we obtain the sharp upper bound for of a general bipartite graph and a series of extremal graphs. Additionally, we give a proper -coloring for a connected bipartite graph having and a dominating cycle...
A graph is called improperly -colorable if the vertex set can be partitioned into subsets such that the graph induced by the vertices of has maximum degree at most for all . In this paper, we mainly study the improper coloring of -planar graphs and show that -planar graphs with girth at least are -colorable.
We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate to the quasi-neutral...
We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate to the quasi-neutral...
We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal...
We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal...
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