An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\colon E\to \mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\mathbb{Z}$. The interval thickness $\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \frac{\log n}{\log \log n} \leq \theta(n) \leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan. In the second part of the paper, we confirm a conjecture of Axenovich that the maximum number of colours used in an interval colouring of a planar graph on $n$ vertices is at most $3n/2-2$.

On interval colourings of graphs

Wearable plant sensors hold tremendous potential for smart agriculture. We report a lower leaf surface-attached multimodal wearable sensor for continuous monitoring of plant physiology by tracking both biochemical and biophysical signals of the plant and its microenvironment. Sensors for detecting volatile organic compounds (VOCs), temperature, and humidity are integrated into a single platform. The abaxial leaf attachment position is selected on the basis of the stomata density to improve the sensor signal strength. This versatile platform enables various stress monitoring applications, ranging from tracking plant water loss to early detection of plant pathogens. A machine learning model was also developed to analyze multichannel sensor data for quantitative detection of tomato spotted wilt virus as early as 4 days after inoculation. The model also evaluates different sensor combinations for early disease detection and predicts that minimally three sensors are required including the VOC sensors.

https://www.science.org/doi/pdf/10.1126/sciadv.ade2232?download=true

Abaxial leaf surface-mounted multimodal wearable sensor for continuous plant physiology monitoring

Stretchable ultra‐narrow (e.g., 10 µm in width) microelectrodes are crucial for the electrophysiological monitoring of single cells providing the fundamental understanding to the working mechanism of neuro network or other electrically functional cells. Current fabrication strategies either focus on the preparation of normal stretchable electrodes with hundreds of micrometers or millimeters in width by using inorganic conductive materials or develop conductive organic polymer gel for ultra‐narrow electrodes which suffer from low stretchability and instability for long‐term implantation, therefore, it is still highly desirable to explore bio‐interfacial ultra‐narrow stretchable inorganic electrodes. Herein, a hybrid strategy is reported to prepare ultra‐narrow multi‐channel stretchable microelectrodes without using photolithography or laser‐assisting etching. A 10 µm × 10 µm monitoring window is fabricated with enhanced interfacial impedance by the special rough surface. The stretchability achieves to 120% for this 10 µm‐width stretchable electrode. Supported by these superior properties, it is demonstrated that the stretchable microelectrodes can detect electrophysiological signals of single cells in vitro and collect electrophysiological signals more precisely in vivo. The reported strategy will open up the accessible preparation of the fine‐size stretchable microelectrode. It will significantly improve the resolution of monitoring and stimulation of inorganic stretchable electrodes.

A Hybrid Strategy‐Based Ultra‐Narrow Stretchable Microelectrodes with Cell‐Level Resolution

A graph is said to be interval colourable if it admits a proper edge-colouring using palette N in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph G is the minimum k such that G can be edge-decomposed into k interval colourable graphs. We show that θ ( n ) , the maximum interval colouring thickness of an n -vertex graph, satisﬁes log( n ) 1 / 3 − o (1) 6 θ ( n ) 6 n 5 / 6+ o (1) , which improves on the trivial lower bound and an upper bound of Axenovich and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan.

A note on the interval colouring thickness of graphs

Silver nanowires (AgNWs) are important materials for flexible transparent electrodes (FTEs). However, the loose stacking of nanowire junctions greatly affects the electric conductivity across adjacent nanowires. Soldering can effectively reduce the wire-wire contact resistance of AgNWs by epitaxially depositing nanosolders at the junctions, but the process normally needs to be performed with high energy consumption. In this work, we proposed a simple room-temperature method to achieve precise welding of junctions by adjusting the wettability of the soldered precursor solution on the surfaces of AgNWs. The nanoscale welding at nanowire cross junctions forms efficient conductive networks. Furthermore, reduced graphene oxide (rGO) was used to improve the stability of FTEs by wrapping the rGO around the AgNW surface. The obtained FTE shows a figure-of-merit (FoM) of up to 439.3 (6.5 Ω/sq at a transmittance of 88%) and has significant bending stability and environmental and acidic stability. A flexible transparent heater was successfully constructed, which could reach up to 160 °C within a short response time (43 s) and exhibit excellent switching stability. When laminating this FTE onto half perovskite solar cells as the top electrodes, the obtained double-side devices achieved power conversion efficiencies as high as 16.15% and 13.91% from each side, pointing out a convenient method for fabricating double-sided photovoltaic devices.

Chemically Welding Silver Nanowires toward Transferable and Flexible Transparent Electrodes in Heaters and Double-Sided Perovskite Solar Cells.

Soft electronics using metal nanowires have attracted notable attention attributed to their high electrical conductivity and mechanical flexibility. However, high-resolution complex patterning of metal nanowires on curvilinear substrates remains a challenge. Here, a micromolding-based method is reported for scalable printing of metal nanowires, which enables complex and highly conductive patterns on soft curvilinear and uneven substrates with high resolution and uniformity. Printing resolution of 20 μm and conductivity of the printed patterns of ~6.3 × 106 S/m are achieved. Printing of grid structures with uniform thickness for transparent conductive electrodes (TCEs) and direct printing of pressure sensors on curved surfaces such as glove and contact lens are also realized. The printed hybrid soft TCEs and smart contact lens show promising applications in optoelectronic devices and personal health monitoring, respectively. This printing method can be extended to other nanomaterials for large-scale printing of high-performance soft electronics.

Curvilinear soft electronics by micromolding of metal nanowires in capillaries

A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs edge-decomposing G. We prove that s(G)=o(n) for any graph G on n vertices. This improves the previously known bound of 2n/5 by Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with interval thickness strictly greater than 2, we construct bipartite graphs whose interval spectrum has arbitrarily many arbitrarily large gaps. Here, an interval spectrum of a graph is the set of all integers t such that the graph has an interval coloring using t colors. Interval colorings of bipartite graphs naturally correspond to no-wait schedules, say for parent-teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with $n$ participants can be coordinated in o(n) no-wait periods. In addition, we show that for any integers t and T, t<T, there is a set of pairs of parents and teachers wanting to talk to each other, such that any no-wait schedules are unstable -- they could last t hours and could last T hours, but there is no possible no-wait schedule lasting x hours if t<x<T.

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/jgt.23003

Michael Zheng

Papers

Curvilinear soft electronics by micromolding of metal nanowires in capillaries

Interval colorings of graphs—Coordinated and unstable no‐wait schedules