Note on minimally kk-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.Comment: 8 page

Dynamics and correlation length scales of a glass-forming liquid in quiescent and sheared conditions

We numerically study dynamics and correlation length scales of a colloidal liquid in both quiescent and sheared conditions to further understand the origin of slow dynamics and dynamic heterogeneity in glass-forming systems. The simulation is performed in a weakly frustrated two-dimensional liquid, where locally preferred order is allowed to develop with increasing density. The four-point density correlations and bond-orientation correlations, which have been frequently used to capture dynamic and static length scales $\xi$ in a quiescent condition, can be readily extended to a system under steady shear in this case. In the absence of shear, we confirmed the previous findings that the dynamic slowing down accompanies the development of dynamic heterogeneity. The dynamic and static length scales increase with $\alpha$-relaxation time $\tau_{\alpha}$ as power-law $\xi\sim\tau_{\alpha}^{\mu}$ with $\mu>0$. In the presence of shear, both viscosity and $\tau_{\alpha}$ have power-law dependence on shear rate in the marked shear thinning regime. However, dependence of correlation lengths cannot be described by power laws in the same regime. Furthermore, the relation $\xi\sim\tau_{\alpha}^{\mu}$ between length scales and dynamics holds for not too strong shear where thermal fluctuations and external forces are both important in determining the properties of dense liquids. Thus, our results demonstrate a link between slow dynamics and structure in glass-forming liquids even under nonequilibrium conditions.Comment: 9 pages, 17 figures. Accepted by J. Phys.: Condens. Matte

Structure, compressibility factor and dynamics of highly size-asymmetric binary hard-disk liquids

By using event-driven molecular dynamics simulation, we investigate effects of varying the area fraction of the smaller component on structure, compressibility factor and dynamics of the highly size-asymmetric binary hard-disk liquids. We find that the static pair correlations of the large disks are only weakly perturbed by adding small disks. The higher-order static correlations of the large disks, by contrast, can be strongly affected. The compressibility factor of the system first decreases and then increases upon increasing the area fraction of the small disks and separating different contributions to it allows to rationalize this non-monotonic phenomenon. Furthermore, adding small disks can influence dynamics of the system in quantitative and qualitative ways. For the large disks, the structural relaxation time increases monotonically with increasing the area fraction of the small disks at low and moderate area fractions of the large disks. In particular, "reentrant" behavior appears at sufficiently high area fractions of the large disks, strongly resembling the reentrant glass transition in short-ranged attractive colloids and the inverted glass transition in binary hard spheres with large size disparity. By tuning the area fraction of the small disks, relaxation process for the small disks shows concave-to-convex crossover and logarithmic decay behavior, as found in other binary mixtures with large size disparity. Moreover, diffusion of both species is suppressed by adding small disks. Long-time diffusion for the small disks shows power-law-like behavior at sufficiently high area fractions of the small disks, which implies precursors of a glass transition for the large disks and a localization transition for the small disks. Therefore, our results demonstrate the generic dynamic features in highly size-asymmetric binary mixtures.Comment: 9 pages, 12 figure