A note on 3-choosability of planar graphs without certain cycles

AbstractSteinberg asked whether every planar graph without 4 and 5 cycles is 3-colorable. Borodin, and independently Sanders and Zhao, showed that every planar graph without any cycle of length between 4 and 9 is 3-colorable. We improve this result by showing that every planar graph without any cycle of length 4, 5, 6, or 9 is 3-choosable

A transdiagnostic data-driven study of children's behaviour and the functional connectome.

Behavioural difficulties are seen as hallmarks of many neurodevelopmental conditions. Differences in functional brain organisation have been observed in these conditions, but little is known about how they are related to a child's profile of behavioural difficulties. We investigated whether behavioural difficulties are associated with how the brain is functionally organised in an intentionally heterogeneous and transdiagnostic sample of 957 children aged 5-15. We used consensus community detection to derive data-driven profiles of behavioural difficulties and constructed functional connectomes from a subset of 238 children with resting-state functional Magnetic Resonance Imaging (fMRI) data. We identified three distinct profiles of behaviour that were characterised by principal difficulties with hot executive function, cool executive function, and learning. Global organisation of the functional connectome did not differ between the groups, but multivariate patterns of connectivity at the level of Intrinsic Connectivity Networks (ICNs), nodes, and hubs significantly predicted group membership in held-out data. Fronto-parietal connector hubs were under-connected in all groups relative to a comparison sample and children with hot vs cool executive function difficulties were distinguished by connectivity in ICNs associated with cognitive control, emotion processing, and social cognition. This demonstrates both general and specific neurodevelopmental risk factors in the functional connectome

Electrical and Electro-Optical Biosensors

Electrical and electro-optical biosensing technologies are critical to the development of innovative POCT devices, which can be used by both professional and untrained personnel for the provision of necessary health information within a short time for medical decisions to be determined, being especially important in an era of global pandemics. This Special Issue includes a few pioneering works concerning biosensors utilizing electrochemical impedance, localized surface plasmon resonance, and the bioelectricity of sensing materials in which the amount of analyte is pertinent to the signal response. The presented results demonstrate the potential of these label-free biosensing approaches in the detection of disease-related small-molecule metabolites, proteins, and whole-cell entities

Terahertz Technology and Its Applications

The Terahertz frequency range (0.1 – 10)THz has demonstrated to provide many opportunities in prominent research fields such as high-speed communications, biomedicine, sensing, and imaging. This spectral range, lying between electronics and photonics, has been historically known as “terahertz gap” because of the lack of experimental as well as fabrication technologies. However, many efforts are now being carried out worldwide in order improve technology working at this frequency range. This book represents a mechanism to highlight some of the work being done within this range of the electromagnetic spectrum. The topics covered include non-destructive testing, teraherz imaging and sensing, among others

Coloración en triangulaciones

Some of the most studied problems in Graph Theory are those referring to the coloring of the graph, being one of the most famous the Three Color Problem. A color set D for a graph G is said to be a 3-coloring if adjacent vertex has a different color of D making the graph 3-coloreable. It seems to be obvious to wonder which graphs are 3-coloreable. Nevertheless, the problem of finding sufficient conditions for a graph to be 3-coloreable in a general graph has been shown by L. Stockmayer in 1979 in his book “Planar 3-colorability is polynomial complete" to be NP-complete. That is why different bounds for x(G) are studied and stated for both arbitrary graphs and for those with a particular structure. Nonetheless, the interest in this parameter is not only to establish new bounds, but also once the bounds have been obtained, either upper or bottom, this naturally brings us the question of knowing if there exists any graph which verifies the equality. Throughout these months, the results achieved about the 3-coloring problem for arbitrary graphs have been studied and, specifically, those results referring to the variants of the 3-coloration problem attending to the sum of colors, the distance between vertex or the parity among the apparition of certain color. This research has been performed not only from a combinatorial point of view but also from an algorithmic point of view and has been restricted to a particular kind of graph, known as maximal outerplanar graphs and denoted by its acronym as MOP's, graph of high importance in both the field of chemistry and polygon triangulations. This project has a double purpose: on the one hand, it seeks to collect those results in the literature which have been observed to be more significant in a review paper or sur- vey; on the other hand, it seeks to established tight combinatorial bounds for some variants of the 3-coloration concept for any n-vertex maximal outerplanar graph. Thus, as main contributions, we will prove several new tight combinatorial bounds for the following variants of coloration concept attending to the sum of the colors been used: sum-coloring, as well as the following variants attending to the existence of a rainbow path: rainbow coloring.---ABSTRACT---Algunos de los problemas más estudiados en Teoría de Grafos son aquellos problemas que hacen referencia a la coloración del mismo, siendo uno de los más clásicos el problema de los Tres Colores. Un conjunto D de colores de un grafo G se dice que es una 3-coloración si vértices adyacentes tienen un color distinto de D haciendo el grafo 3-coloreable. Parece entonces obvio preguntarse qué grafos son 3-coloreables. Sin embargo, ya en 1979 L. Stockmayer en su artículo “Planar 3-colorability is polynomial complete" probó que este problema es NP-completo. Es por ello por lo que se estudian y establecen cotas para x(G) para el caso de grafos cualesquiera o para grafos con cierta estructura. Sin embargo, el interés en este parámetro no sólo radica en establecer una cota, sino que una vez obtenida dicha cota, ya sea superior o inferior, quedaría comprobar la existencia de algún grafo que verifique la igualdad. A lo largo de estos meses de trabajo, se han estudiado los resultados obtenidos hasta la fecha en el problema de la 3-coloración de grafos en general y más concretamente sobre aquellas variantes de 3-coloración que atienden a la suma de los colores, la distancia entre vértices o la paridad en la aparición de cierto color. Este estudio se ha llevado a cabo tanto desde el punto de vista combinatorio como algorítmico y se ha restringido a un tipo particular de grafos, conocidos como grafos periplanos maximales y de nominados a partir de ahora por sus siglas en inglés MOP's (maximal outerplanar graphs), grafos de gran importancia tanto en el ámbito de la química como en el de triangulaciones de polígonos. Con este proyecto se persigue un doble objetivo: por un lado, se pretende recopilar aque- llos resultados más significativos de la bibliografía en un artículo de tipo "survey"; por otro, obtener nuevos resultados sobre variantes de dominación para MOP's. Así, como aporte de nuestro trabajo, probaremos nuevas cotas que se han establecido tanto para los criterios de 3-coloración que atienden a la suma de colores utilizados en la coloración: sum-coloring, como para variantes que atienden a la existencia de caminos irisados en la coloración del grafo: coloración irisada