Class two 1-planar graphs with maximum degree six or seven

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.Comment: 3 pages, 2 figure

Probing the interaction between dark energy and dark matter with the parametrized post-Friedmann approach

There possibly exists some direct, non-gravitational coupling between dark energy and dark matter. This possibility should be seriously tested by using observations, which requires us to understand such a scenario from the aspects of both expansion history and growth of structure. It is found that once calculating the perturbations in the interacting dark energy (IDE) scenario, for most cases the curvature perturbation on superhorizon scales is divergent, which is a catastrophe for the IDE cosmology. We found a solution to this issue, which is to establish an effective theory to treat the dark energy perturbations totally based on the basic facts of dark energy. This scheme generalizes the parametrized post-Friedmann framework of uncoupled dark energy and can be used to cure the instability of the IDE cosmology. The whole parameter space of IDE models can henceforward be explored by observational data. The IDE scenario can thus be tested or falsified with current and future observational data by using the PPF approach. We expect that the future highly accurate observational data would offer the certain answer to the question whether there is a direct coupling between dark energy and dark matter.Comment: 3 pages, 2 figure

Weighing neutrinos in dynamical dark energy models

We briefly review the recent results of constraining neutrino mass in dynamical dark energy models using cosmological observations and summarize the findings. (i) In dynamical dark energy models, compared to $\Lambda$CDM, the upper limit of $\sum m_\nu$ can become larger and can also become smaller. In the cases of phantom and early phantom (i.e., the quintom evolving from $w<-1$ to $w>-1$), the constraint on $\sum m_\nu$ becomes looser; but in the cases of quintessence and early quintessence (i.e., the quintom evolving from $w>-1$ to $w<-1$), the constraint on $\sum m_\nu$ becomes tighter. (ii) In the holographic dark energy (HDE) model, the tightest constraint on $\sum m_\nu$, i.e., $\sum m_\nu<0.105$ eV, is obtained, which is almost equal to the lower limit of $\sum m_\nu$ of IH case. Thus, it is of great interest to find that the future neutrino oscillation experiments would potentially offer a possible falsifying scheme for the HDE model. (iii) The mass splitting of neutrinos can influence the cosmological fits. We find that the NH case fits the current observations slightly better than the IH case, although the difference of $\chi^2$ of the two cases is still not significant enough to definitely distinguish the neutrino mass hierarchy.Comment: 3 page

The edge chromatic number of outer-1-planar graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs