29,253 research outputs found

### Note on minimally $k$-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

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