Packing chromatic vertex-critical graphs

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical

Processing eutectics in space

Experimental work is reported which was directed toward obtaining interface shape control while a numerical thermal analysis program was being made operational. An experimental system was developed in which the solid-liquid interface in a directionally solidified aluminum-nickel eutectic could be made either concave to the melt or convex to the melt. This experimental system provides control over the solid-liquid interface shape and can be used to study the effect of such control on the microstructure. The SINDA thermal analysis program, obtained from Marshall Space Flight Center, was used to evaluate experimental directional solidification systems for the aluminum-nickel and the aluminum-copper eutectics. This program was applied to a three-dimensional ingot, and was used to calculate the thermal profiles in axisymmetric heat flow. The results show that solid-liquid interface shape control can be attained with physically realizable thermal configurations and the magnitudes of the required thermal inputs were indicated

Hypnotic Tactile Anesthesia: Psychophysical and Signal-Detection Analyses.

Two experiments that studied the effects of hypnotic suggestions on tactile sensitivity are reported. Experiment 1 found that suggestions for anesthesia, as measured by both traditional psychophysical methods and signal-detection procedures, were linearly related to hypnotizability. Experiment 2 employed the same methodologies in an application of the real-simulator paradigm to examine the effects of suggestions for both anesthesia and hyperesthesia. Significant effects of hypnotic suggestion on both sensitivity and bias were found in the anesthesia condition but not for the hyperesthesia condition. A new bias parameter, C', indicated that much of the bias found in the initial analyses was artifactual, a function of changes in sensitivity across conditions. There were no behavioral differences between reals and simulators in any of the conditions, though analyses of postexperimental interviews suggested the 2 groups had very different phenomenal experiences