The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.

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Faculty Opinions recommendation of A topoisomerase II-dependent mechanism for resetting replicons at the S-M-phase transition.

Faculty Opinions – Post-Publication Peer Review of the Biomedical Literature ◽

10.3410/f.1112214.568207 ◽

2008 ◽

Author(s):

Stephen Kearsey

Keyword(s):

Phase Transition ◽

Topoisomerase Ii ◽

M Phase ◽

Dependent Mechanism

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Time Dependent Lyotropic Chromonic Textures in Microfluidic Confinements

Crystals ◽

10.3390/cryst11010035 ◽

2020 ◽

Vol 11 (1) ◽

pp. 35 ◽

Cited By ~ 1

Author(s):

Anshul Sharma ◽

Irvine Lian Hao Ong ◽

Anupam Sengupta

Keyword(s):

Phase Transition ◽

Aspect Ratio ◽

Temporal Dynamics ◽

Initial Conditions ◽

Flow Behavior ◽

Medical Diagnostics ◽

Time Dependent ◽

Disodium Cromoglycate ◽

M Phase ◽

Static Conditions

Nematic and columnar phases of lyotropic chromonic liquid crystals (LCLCs) have been long studied for their fundamental and applied prospects in material science and medical diagnostics. LCLC phases represent different self-assembled states of disc-shaped molecules, held together by noncovalent interactions that lead to highly sensitive concentration and temperature dependent properties. Yet, microscale insights into confined LCLCs, specifically in the context of confinement geometry and surface properties, are lacking. Here, we report the emergence of time dependent textures in static disodium cromoglycate (DSCG) solutions, confined in PDMS-based microfluidic devices. We use a combination of soft lithography, surface characterization, and polarized optical imaging to generate and analyze the confinement-induced LCLC textures and demonstrate that over time, herringbone and spherulite textures emerge due to spontaneous nematic (N) to columnar M-phase transition, propagating from the LCLC-PDMS interface into the LCLC bulk. By varying the confinement geometry, anchoring conditions, and the initial DSCG concentration, we can systematically tune the temporal dynamics of the N- to M-phase transition and textural behavior of the confined LCLC. Overall, the time taken to change from nematic to the characteristic M-phase textures decreased as the confinement aspect ratio (width/depth) increased. For a given aspect ratio, the transition to the M-phase was generally faster in degenerate planar confinements, relative to the transition in homeotropic confinements. Since the static molecular states register the initial conditions for LC flows, the time dependent textures reported here suggest that the surface and confinement effects—even under static conditions—could be central in understanding the flow behavior of LCLCs and the associated transport properties of this versatile material.

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Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

The Electronic Journal of Combinatorics ◽

10.37236/7353 ◽

2019 ◽

Vol 26 (2) ◽

Author(s):

Jinko Kanno ◽

Songling Shan

Keyword(s):

Maximum Degree ◽

Hamiltonian Cycle ◽

Simple Graph ◽

Chromatic Index ◽

New Techniques ◽

Critical Graph ◽

Degree Conditions ◽

Delta G ◽

Proper Subgraph

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

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A topoisomerase II-dependent mechanism for resetting replicons at the S-M-phase transition

Genes & Development ◽

10.1101/gad.445108 ◽

2008 ◽

Vol 22 (7) ◽

pp. 860-865 ◽

Cited By ~ 29

Author(s):

O. Cuvier ◽

S. Stanojcic ◽

J.-M. Lemaitre ◽

M. Mechali

Keyword(s):

Phase Transition ◽

Topoisomerase Ii ◽

M Phase ◽

Dependent Mechanism

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The Polo-like kinase Plx1 is a component of the MPF amplification loop at the G2/M-phase transition of the cell cycle in Xenopus eggs

Journal of Cell Science ◽

10.1242/jcs.111.12.1751 ◽

1998 ◽

Vol 111 (12) ◽

pp. 1751-1757 ◽

Cited By ~ 16

Author(s):

A. Abrieu ◽

T. Brassac ◽

S. Galas ◽

D. Fisher ◽

J.C. Labbe ◽

...

Keyword(s):

Phase Transition ◽

Cell Cycle ◽

Cyclin A ◽

Cyclin B ◽

Cdc2 Kinase ◽

C Terminus ◽

M Phase ◽

Egg Extracts ◽

Amplification Loop

We have investigated whether Plx1, a kinase recently shown to phosphorylate cdc25c in vitro, is required for activation of cdc25c at the G2/M-phase transition of the cell cycle in Xenopus. Using immunodepletion or the mere addition of an antibody against the C terminus of Plx1, which suppressed its activation (not its activity) at G2/M, we show that Plx1 activity is required for activation of cyclin B-cdc2 kinase in both interphase egg extracts receiving recombinant cyclin B, and cycling extracts that spontaneously oscillate between interphase and mitosis. Furthermore, a positive feedback loop allows cyclin B-cdc2 kinase to activate Plx1 at the G2/M-phase transition. In contrast, activation of cyclin A-cdc2 kinase does not require Plx1 activity, and cyclin A-cdc2 kinase fails to activate Plx1 and its consequence, cdc25c activation in cycling extracts.

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Extremal trees with respect to the Steiner Wiener index

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500678 ◽

2019 ◽

Vol 11 (06) ◽

pp. 1950067

Author(s):

Jie Zhang ◽

Guang-Jun Zhang ◽

Hua Wang ◽

Xiao-Dong Zhang

Keyword(s):

Maximum Degree ◽

Computational Analysis ◽

Degree Sequence ◽

Wiener Index ◽

Extremal Problems ◽

Difficult Problem ◽

Computational Results ◽

Number Of Leaves ◽

Extremal Structure ◽

Chemical Trees

The well-known Wiener index is defined as the sum of pairwise distances between vertices. Extremal problems with respect to it have been extensively studied for trees. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of minimum sizes of subgraphs that span [Formula: see text] given vertices over all possible choices of the [Formula: see text] vertices. We consider the extremal problems with respect to the Steiner Wiener index among trees of a given degree sequence. First, it is pointed out minimizing the Steiner Wiener index in general may be a difficult problem, although the extremal structure may very likely be the same as that for the regular Wiener index. We then consider the upper bound of the general Steiner Wiener index among trees of a given degree sequence and study the corresponding extremal trees. With these findings, some further discussion and computational analysis are presented for chemical trees. We also propose a conjecture based on the computational results. In addition, we identify the extremal trees that maximize the Steiner Wiener index among trees with a given maximum degree or number of leaves.

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