Common graphs with arbitrary connectivity and chromatic number

A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. We prove that, given $k,r>0$, there exists a $k$-connected common graph with chromatic number at least $r$. The result is built upon the recent breakthrough of Kr\'a\v{l}, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.Comment: 6 page

A characterization of graphs of radius-rr flip-width at most 22

The $r$-flip-width of a graph, for $r\in \mathbb{N}\cup \{\infty\}$, is a graph parameter defined in terms of a variant of the cops and robber game, called a flipper game, and it was introduced by Toru\'{n}czyk [Flip-width: Cops and robber on dense graphs, arXiv:2302.00352]. We prove that for every $r\in (\mathbb{N}\setminus \{1\})\cup \{\infty\}$, the class of graphs of $r$-flip-width at most $2$ is exactly the class of ($C_5$, bull, gem, co-gem)-free graphs, which are known as totally decomposable graphs with respect to bi-joins.Comment: 14 pages, 1 figur

Development of an anti-pollution coating process technology for the application of an on-site PV module

This study aimed to apply annealing processes during the coating of photovoltaic (PV) module glasses to PV modules already installed through an easy and simple procedure. Three types of annealing treatments were applied to PV module glasses, i.e., furnace, rapid thermal annealing (RTA) and torch. Among these, torch annealing, which can be easily carried out at PV module installation sites, was applied to PV module glasses using different numbers of repetition. Light transmittance, contact angle, anti-pollution characteristics, adhesion and hardness of the functional coating films after using different annealing treatment times and methods were measured, and it was confirmed that these characteristics varied depending on the annealing treatment times and methods. Through this, it was possible to optimize the process conditions that provide excellent anti-pollution characteristics and could be easily utilized at on-site PV modules