Three classes of space-time with a uniformly accelerating and nonrotating gravitational source

Three classes of Petrov type-D nonrotating empty spaces, which we denote asA ₊ ^* ,A ₋ ^* , anda ₀ ^* , can be seen to represent, respectively, a gravitational field of a uniformly accelerating source which moves with velocity slower, faster than, or equal to that of light. The first two classes of space-time,A ₊ ^* andA ₋ ^* , approach to the last one,a ₀ ^* , as the acceleration becomes greater and greater. The inertial frame in a flat space-time, relative to which a faster-thanlight particle moving along a spacelike geodesic behaves as an ordinary particle, is given and discussed. The behaviors of Killing horizons in the classes,A ₊ ^* andA ₋ ^* are investigated. Comparisons are made between the analysis by Farhoosh and Zimmerman and ours. Our identification of the acceleration parameter seems to be more appropriate than that by the above authors.     

Half-integral packing of odd cycles through prescribed vertices

The well-known theorem of Erd?s-Pósa says that a graph G has either k disjoint cycles or a vertex set X of order at most f(k) for some function f such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erd?s-Pósa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erd?s-Pósa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erd?s-Pósa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.