Conjectures on uniquely 3-edge-colorable graphs

A graph $G$ is {\it uniquely k-edge-colorable} if the chromatic index of $G$ is $k$ and every two $k$-edge-colorings of $G$ produce the same partition of $E(G)$ into $k$ independent subsets.For any $k\ne 3$, a uniquely $k$-edge-colorable graph $G$ is completely characterized;$G\cong K_2$ if $k=1$, $G$ is a path or an even cycle if $k=2$,and $G$ is a star $K_{1,k}$ if $k\geq 4$.On the other hand, there are infinitely many uniquely 3-edge-colorable graphs, and hence, there are many conjectures for the characterization of uniquely 3-edge-colorable graphs.In this paper, we introduce a new conjecture which connects conjectures of uniquely 3-edge-colorable planar graphs with those of uniquely 3-edge-colorable non-planar graphs

Characterization of outerplanar graphs with equal 2-domination and domination numbers

A {\em $k$-domination number} of a graph $G$ is minimum cardinality of a $k$-dominating set of $G$, where a subset $S \subseteq V(G)$ is a {\em $k$-dominating set} if each vertex $v\in V(G)\setminus S$ is adjacent to at least $k$ vertices in $S$. It is known that for any graph $G$ with $\Delta(G) \geq k \geq 2$, $\gamma_k(G) \geq \gamma(G) + k - 2$, and then $\gamma_k(G) \u3e \gamma(G)$ for any $k\geq 3$, where $\gamma(G) = \gamma_1(G)$ is the usual domination number. Thus, it is the most interesting problem to characterize graphs $G$ with $\gamma_2(G) = \gamma(G)$. In this paper, we characterize outerplanar graphs with equal 2-domination and domination numbers

Magnetohydrodynamic shocks in and above post-flare loops: two-dimensional simulation and a simplified model

Solar flares are an explosive phenomenon, where super-sonic flows and shocks are expected in and above the post-flare loops. To understand the dynamics of post-flare loops, a two-dimensional magnetohydrodynamic (2D MHD) simulation of a solar flare has been carried out. We found new shock structures in and above the post-flare loops, which were not resolved in the previous work by Yokoyama and Shibata 2001. To study the dynamics of flows along the reconnected magnetic field, kinematics and energetics of the plasma are investigated along selected field lines. It is found that shocks are crucial to determine the thermal and flow structures in the post-flare loops. On the basis of the 2D MHD simulation, we have developed a new post-flare loop model which we call the pseudo-2D MHD model. The model is based on the 1D MHD equations, where all the variables depend on one space dimension and all the three components of the magnetic and velocity fields are considered. Our pseudo-2D model includes many features of the multi-dimensional MHD processes related to magnetic reconnection (particularly MHD shocks), which the previous 1D hydrodynamic models are not able to include. We compare the shock formation and energetics of a specific field line in the 2D calculation with those in our pseudo-2D MHD model, and we found that they give similar results. This model will allow us to study the evolution of the post-flare loops in a wide parameter space without expensive computational cost and without neglecting important physics associated with magnetic reconnection.Comment: 51 pages, 22 figures. Accepted by Ap

Facial Achromatic Number of Triangulations with Given Guarding Number

A (not necessarily proper) $k$-coloring $c : V(G) \rightarrow \{1,2,\dots,k\}$ of a graph $G$ on a surface is a {\em facial $t$-complete $k$-coloring} if every $t$-tuple of colors appears on the boundary of some face of $G$. The maximum number $k$ such that $G$ has a facial $t$-complete $k$-coloring is called a {\em facial $t$-achromatic number} of $G$, denoted by $\psi_t(G)$. In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a {\em guarding number} of a graph $G$ embedded on a surface, denoted by \gd(G), is the smallest size of its {\em guarding set} which is a generalized concept of guards in the art gallery problem. We show that for any graph $G$ embedded on a surface, \psi_{\Delta(G^*)}(G) \leq \gd(G) + \Delta(G^*) - 1, where $\Delta(G^*)$ is the largest face size of $G$. Furthermore, we investigate sufficient conditions for a triangulation $G$ on a surface to satisfy \psi_{3}(G) = \gd(G) + 2. In particular, we prove that every triangulation $G$ on the sphere with \gd(G) = 2 satisfies the above equality and that for one with guarding number $3$, it also satisfies the above equality with sufficiently large number of vertices

Preparing mechanical squeezing of a macroscopic pendulum near quantum regimes

We present the mechanical squeezing of a mg-scale suspended mirror (i.e. a pendulum) near quantum regimes through continuous linear position measurement. The experiment involved the pendulum interacting with photon coherent fields in a detuned optical cavity. The position uncertainty in the measured data is reduced and squeezed to $470$ times the zero-point amplitude $x_{\rm zpf}$ with a purity of about 0.0004, by means of optimal state estimation through causal Wiener filtering. The purity of the squeezed state is clearly maximized by the Wiener filter, based on precisely identified optomechanical parameters. This is the first step for measurement-based quantum control of macroscopic pendulums, e.g. generation of an entanglement state between macroscopic pendulums. Such quantum control will provide a direct insight into the quantum to classical transition and will pave the way to test semiclassical gravity and gravity sourced by macroscopic quantum oscillators.Comment: 6 pages, 3figures, and a supplemental materia

Facial achromatic number of triangulations on the sphere (Women in Mathematics)

A graph consists of a set of vertices and a set of edges. A coloring of a graph is an assigning of colors to the vertices such that any adjacent vertices receive different colors. In particular, a coloring is called complete if every pair of colors appear on some edge. In this talk, we expand complete colorings of graphs to those of graphs embedded on surfaces and consider such colorings of even triangulations on the sphere