An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Development of Semiconducting Tin Dinitridocarbonate Compounds by Solid-State Metathesis Reactions

This work focuses on solid-state syntheses of tin compounds and some of their properties. Most of these compounds were obtained by solid-state metathesis (SSM) reactions between lithium carbodiimide and tin halides. Furthermore, some halides and oxide halides have been synthesized. The majority of the described compounds have been characterized by single-crystal X-ray diffraction (XRD). In the reaction of Li2(CN2) and SnCl2, Sn(CN2) was produced, along with Sn2O(CN2), as a byproduct. Further research has led to the discovery of Sn4Cl2(CN2)3 and Sn9O5Cl4(CN2)2. Moreover, Sn4Cl2(CN2)3 was shown to act as a precursor compound in the syntheses of Sn(CN2), Sn2O(CN2), and Sn9O5Cl4(CN2)2. Sn4Cl2(CN2)3 is formed in these syntheses at temperatures below 200 °C and transforms into either Sn(CN2), Sn2O(CN2), or Sn9O5Cl4(CN2)2 at temperatures above 300 °C. CuWO4-based photoanodes mixed with Sn2O(CN2) showed improved efficiencies compared to pure CuWO4 electrodes. Some tin carbodiimides were investigated by density functional theory (DFT) calculations, revealing their electronic structures, and indicating them to be semiconductors with band gaps on the order of 1 to 3 eV. The divalent tin carbodiimide compounds Sn4Cl2(CN2)3, Sn9O5Cl4(CN2)2, and Sn2O(CN2) were studied by Mössbauer spectroscopy to identify possible traces of Sn4+. No tin IV could be detected in samples of Sn4Cl2(CN2)3; however, samples of Sn9O5Cl4(CN2)2 and Sn2O(CN2) showed the presence of tetravalent tin. This Sn4+ presumably originates from the starting material SnO, which has also been shown to contain tin in the oxidation state IV. Analogous reactions with Li2(CN2) and SnBr2 (instead of SnCl2) led to the discovery of two compounds, Sn4Br2(CN2)3 and LiSn2Br3(CN2). Sn4Br2(CN2)3 is isotypic to Sn4Cl2(CN2)3, whereas LiSn2Br3(CN2) is isotypic to the mineral bideauxite (AgPb2Cl3(F,OH)2). LiSn2Br3(CN2) is thought to show negative thermal expansion at temperatures below 25 K, as indicated by density functional perturbation theory (DFPT) calculations. Reactions between tetravalent tin halides and lithium or sodium carbodiimide led to the formation of already known ternary compounds containing either Li or Na, e.g., A2Sn(CN2)3, A = Li, Na. To avoid the formation of ternary Li/Na containing dinitridocarbonates, reactions of Pb(CN2) and SnBr4 were carried out, leading to the formation of yellow single crystals, that were identified as Pb14.66Sn7.34Br26(CN2)7O2. VIII The compound containing only divalent tin has a complex crystal structure, where one metal site is occupied by both tin and lead. The tin to lead ratio could be determined by energy-dispersive X-ray spectroscopy (EDX) and is consistent with the crystal structure refinement. In course of explorative syntheses in the system of Li2(CN2), SnO, and SnX2, (X = Cl, Br, I), three tin oxide halides, Sn7O4Cl6, Sn7O4Br6, and Sn4OI6, were obtained. Since some tin oxide halides (in particular tin oxide chloride) were mentioned in the literature several decades ago, but without any crystal structure data, the discovery of these three compounds closed a long-standing gap between tin(II) oxide and tin halides. From similar reactions with Li2(CN2) and SnI2, a ternary lithium tin iodide, LiSn3I7, was obtained and later synthesized from SnI2 and LiI. Its crystal structure is closely related to SnI2, but with one shared position of lithium and tin. Recorded 7Li, 119Sn, and 127I solid-state nuclear magnetic resonance (ssNMR) spectra of LiSn3I7 and some related compounds demonstrated the incorporation of Li into the structure and showed the presence of two distinct tin sites. Besides syntheses with tin halides, reactions between lithium carbodiimide and tungsten oxide halides were carried out to synthesize a tungsten oxide carbodiimide. Since reactions with tungsten oxide chlorides and tungsten oxide bromide failed, a tungsten oxide iodide known from textbooks and articles, WO2I2, was synthesized to perform further reactions with it. Single crystals of WO2I2 were synthesized and its previously unknown crystal structure was determined. In addition, a second compound with the composition W2O3I4 was obtained and structurally characterized. WO2I2 was synthesized from a mixture of WO3, W, and I2 at a temperature gradient of 800 to 300 °C. WO2I2 has been reported in the literature to play an important role in chemical transport reactions in tungsten halogen lamps, although its crystal structure was previously unknown. When heated above 400 °C, WO2I2 transforms into W2O3I4, a compound completely unknown in the literature so far