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Abstract: Thin polymer films (TPFs) are indispensable elements in numerous technologies ranging from liquid encapsulation to biotechnology to electronics. However, their production typically relies on wet chemistry involving organic solvents or chemical vapor deposition, necessitating elaborate equipment and often harsh conditions. Here, an eco-friendly, fast, and facile synthesis of water-templated interfacial polymers based on cyanoacrylates (superglues, CAs) that yield thin films with tailored properties is demonstrated. Specifically, by exposing a cationic surfactant-laden water surface to cyanoacrylate vapors, surfactant-modulated anionic polymerization produces a manipulable thin polymer film with a thickness growth rate of 8 nm min−1. Furthermore, the shape and color of the film are precisely controlled by the polymerization kinetics, wetting conditions, and/or exposure to patterned light. Using various interfaces as templates for film growth, including the free surface of drops and soap bubbles, the developed method advantageously enables in situ packaging of chemical and biological cargos in liquid phase as well as the encapsulation of gases within solidified bubbles. Simple, versatile, and biocompatible, this technology constitutes a potent platform for programmable coating and soft/smart encapsulation of fluids.
Keywords: thin polymer films, fluid encapsulation
Published in DiRROS: 07.10.2024; Views: 236; Downloads: 345
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Abstract: By suitably adjusting the tropical algebra technique we compute the rainbow independent domination numbers of several infinite families of graphs including Cartesian products $C_n \Box P_m$ and $C_n \Box C_m$ for all $n$ and $m\le 5$, and generalized Petersen graphs $P(n,2)$ for $n \ge 3$.
Keywords: graph theory, rainbow independent domination number, path algebra, tropical algebra
Published in DiRROS: 03.10.2024; Views: 228; Downloads: 109
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Abstract: In a graph $G$, let $\rho_\triangle(G)$ denote the minimum size of a set of edges and triangles that cover all edges of $G$, and let $\alpha_1(G)$ be the maximum size of an edge set that contains at most one edge from each triangle. Motivated by a question of Erdős, Gallai, and Tuza, we study the relationship between $\rho_\triangle(G)$ and $\alpha_1(G)$ and establish a sharp upper bound on $\rho_\triangle(G)$. We also prove Nordhaus-Gaddum-type inequalities for the considered invariants.
Keywords: edge-disjoint triangles, edge clique covering, Nordhaus-Gaddum inequality
Published in DiRROS: 03.10.2024; Views: 209; Downloads: 112
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Abstract: The Maker-Breaker domination game is played on a graph $G$ by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex $v \in V(G)$. Dominator's fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant $\gamma'_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}(G)$) which is the smallest integer $k$ such that, under any strategy of Dominator, Staller can win the game by playing at most $k$ vertices, if Staller (resp., Dominator) plays first on the graph $G$. We prove some basic properties of $\gamma_{\rm SMB}(G)$ and $\gamma'_{\rm SMB}(G)$ and study the parameters' changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality $\delta(G)+1 \le \gamma'_{\rm SMB}(G) \le \gamma_{\rm SMB}(G)$ always holds and that for every three integers $r,s,t$ with $2\le r\le s\le t$, there exists a graph $G$ such that $\delta(G)+1 = r$, $\gamma'_{\rm SMB}(G) = s$, and $\gamma_{\rm SMB}(G) = t$. We prove exact formulas for $\gamma'_{\rm SMB}(G)$ where $G$ is a path, or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.
Keywords: domination game, Maker–Breaker game, winning number, Maker-Breaker domination game, closed neighborhood hypergraph
Published in DiRROS: 03.10.2024; Views: 229; Downloads: 108
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Abstract: Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. who showed that by $2$-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation ${1 \over 2} - {7 \over 50}$ of the total number of crossings.
Keywords: arrangement of points, flat, hyperplane
Published in DiRROS: 03.10.2024; Views: 188; Downloads: 93
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