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Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, in which two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let $K_{k,n-k}$ be a complete bipartite graph of order $n$. In 1974, Wilson characterized the connectedness of $\mathsf{FS}(K_{1,n-1},Y)$ by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of $\mathsf{FS}(K_{k,n-k},Y)$ for any $Y$ and all $k\ge 2$, including $Y$ being a random graph, as suggested by Defant and Kravitz, and pose some open problems.

A distributed zero-sequence-current (ZSC) carrier signal injection via current transformer (CT) is proposed for underground cable single-phase-to-ground fault location in neutral isolated medium voltage (MV) distribution networks. High-frequency ZSC carrier signal is directly injected to a high-voltage underground cable of an energized distribution network via CT. The injected carrier signal travels along the cable to the receiving center in the substation, and is processed for quick fault line selection and fault location. No external power supplies or signal generators are needed due to the self-contained nature of the injector. The device can be constructed as a sleeve to be mounted around the underground power cable. This permits economical retrofitting to existing installations of power line communication (PLC) devices. In this paper, the operational principle and the application considerations for such an injection device and fault location tricks are explained, followed by the theory behind the proposed technique. Finally, MATLAB based numerical simulations are performed and a low-voltage prototype is tested to verify the validity of the proposed approach.

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k , the Gallai Ramsey number G R k ( H ) is the minimum integer N such that every Gallai k -coloring of K N contains a monochromatic copy of H . A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and l , the l -uniform Ramsey number R k l ( H ) is the minimum integer N such that any uniform k -coloring of any complete multipartite graph on N vertices with each part of cardinality no more than l , contains a monochromatic copy of H . Let K m , n denote a complete bipartite graph. Wu et al. conjectured that G R k ( K m , n ) = ( n − 1 ) ( k − 2 ) + R 2 1 ( K m , n ) if R 2 1 ( K m , n ) ≥ 3 m − 2 , where k ≥ 3 and m ≥ n ≥ 2 . In this paper, we show that G R k ( K m , n ) = ( n − 1 ) ( k − 2 ) + R 2 m − 1 ( K m , n ) for all integers k ≥ 3 and m ≥ n ≥ 2 . Based on this result, we improve the known general bounds for G R k ( K m , n ) , and confirm the conjecture for some complete bipartite graphs.

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k -coloring is a Gallai coloring that uses at most k colors. Given an integer k ≥ 2 , and graphs H 1 , H 2 , … , H k , the Gallai-Ramsey number G R ( H 1 , H 2 , … , H k ) is the least positive integer N such that every Gallai k -coloring of the complete graph K N contains a monochromatic copy of H i in color i for some i ∈ { 1 , 2 , … , k } . Let m G denote the union of m disjoint copies of G . This paper studies the Gallai–Ramsey number G R ( t 1 K 3 , … , t k K 3 ) for all integers t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1 . The case t 1 = t 2 = ⋯ = t k = 1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of G R ( t 1 K 3 , … , t k K 3 ) for all t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1 , and prove that both the upper and lower bounds can be attained for some t i , i ∈ { 1 , … , k } .

For two graphs $$G_1$$ and $$G_2$$ , the star-critical Ramsey number $$r_*(G_1,G_2)$$ is the minimum integer k such that any red/blue edge-coloring of $$K_{r-1}\sqcup K_{1,k}$$ contains a red copy of $$G_1$$ or a blue copy of $$G_2$$ , where r is the classical Ramsey number $$R(G_1,G_2)$$ and $$K_{r-1}\sqcup K_{1,k}$$ is the graph obtained from a $$K_{r-1}$$ and an additional vertex v by joining v to k vertices of $$K_{r-1}$$ . Let $$C_n$$ denote a cycle of order n and $$W_m$$ a wheel of order $$m+1$$ . Hook (2010) proved that $$r_*(C_n,W_3)=2n$$ for $$n\ge 5$$ . In this paper, it is shown that $$r_*(C_n,W_m)=2n$$ for m odd, $$n\ge m\ge 5$$ and $$n\ge 60$$ .

The generalized Turan number $$ex(n,K_s,H)$$ is defined to be the maximum number of copies of a complete graph $$K_s$$ in any H-free graph on n vertices. Let F be an even linear forest, that is, F consists of paths of even order. In this paper, we determine the value of $$ex(n,K_s,F)$$ , which generalizes or improves some known results on generalized Turan number and classical Turan number.

In this article, the random vibration test and finite-element simulation analysis of SAC305 solder joints in BGA packaged electronic chips were carried out. A failure model for BGA solder joints was established based on the life data. And vulnerable parts in electronic chips were analyzed. First, the random vibration test was conducted. Second, a lognormal distribution model of solder joints by means of the life data was established and analyzed based on the fracture mechanics theory. Finally, the finite-element simulation model of BGA packaged electronic chip was established and discussed. Failure characteristics of solder joints at different positions in electronic chips were compared and the failure mode of the solder joint was discussed using the simulation result and the observation result with scanning electron microscope (SEM). The results show that the life of each single solder joint in BGA packaged electronic chips follows a logarithmic normal distribution under the random vibration. Solder balls located at the four outermost corners of the chip are subjected to the greatest stress, which are the weakest parts in electronic chips. Cracks in solder joints are generated near the package side and emerged from the edge of the intermetallic compound (IMC) layer with the largest curvature.

Objective. This study aims to find the novel lncRNAs closely related to the progression of oral squamous cell carcinoma (OSCC) by comprehensively analyzing microarray. Methods. Chip dataset GSE84805 was downloaded from the Gene Expression Omnibus (GEO) database, lncRNA expression profiles of OSCC and paracancerous tissue were obtained, probes sequences reannotation was conducted, and differentially expressed lncRNAs (DELs) and differentially expressed genes (DEGs) were identified. Finally, these data were analyzed by constructing the lncRNA-function enrichment network. Results. We found that 465 lncRNAs are differentially expressed consisting of 193 upregulated lncRNAs and 272 downregulated lncRNAs. Meanwhile, 811 DEGs were identified with 498 upregulated genes and 313 downregulated genes. Analysis of the lncRNA-function enrichment network showed that these aberrant lncRNAs may be related to focal adhesion, inflammatory response pathway, cell cycle, matrix metalloproteinases, and other biological functions. Also, we found that some key lncRNAs such as LINC00152 and HOXA11-AS have been shown to play an important role in tumor proliferation and migration. Conclusion. The key lncRNAs may serve as potential molecular markers or therapeutic targets in OSCC formation and development. It can also help us to understand the molecular mechanism of occurrence and development in OSCC.

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$, of a graph $G$, is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$. Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$. In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$.

Let $$G=(V(G),E(G))$$ be a graph and s, t integers with $$s\le t$$. If we can assign an s-subset $$\phi (v)$$ of the set $$\{1, 2,\ldots ,t\}$$ to each vertex v of V(G) such that $$\phi (u)\cap \phi (v)=\emptyset $$ for every edge $$uv\in E(G)$$, then G is called (t : s)-colorable, and such an assignment $$\phi $$ is called a (t : s)-coloring of G. Let $$C_n$$ denote a cycle of length n. In this paper, we show that every planar graph without $$C_4$$ and $$C_6$$ is (7 : 2)-colorable and thus has fractional chromatic number at most 7/2.

Solenoid valves are important electromagnetic devices which are used widely in various fields. Considering the uncertainty of failure mechanisms in practical operation, a thermal-structure coupled model of a direct action solenoid valve is constructed by using the finite element method. The coupled model provides useful information such as temperature distribution and stress distribution for failure mechanism study of the solenoid valve. The model results predict that the failure of the solenoid valve is closely related to the thermal expansion inside the coil, which means the high temperature generated by thermal expansion melts the insulation layers. Under the combined action of high temperature and coupled stress, the shorting of coils occurs and the resistance decreases which may cause the eventual failure of the solenoid valve. To verify the prediction of the finite element model, a degradation experiment is designed. The results suggest that the failure of the solenoid valve is a gradual degradation process, and there are several mutations of coil temperature and resistance. The high temperature melts the insulation layers and causes the shorting of coils, which further decreases the resistance and increases the current. Later more heat is generated to rise coil temperature and melt insulation layers, which will short out the coils and drop the resistance. With such a relationship, as the temperature rises alternately, the resistance decreases alternately until the solenoid valve completely fails. The model provides a reference for failure mechanism study and life prolonging research of solenoid valves.

Consider two graphs $X$ and $Y$, each with $n$ vertices. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, where two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Alon, Defant, and Kravitz showed that if $X$ and $Y$ are two independent random graphs in $\mathcal{G}(n,p)$, then the threshold probability guaranteeing the connectedness of $\mathsf{FS}(X,Y)$ is $p_0=n^{-1/2+o(1)}$, and suggested to investigate the general asymmetric situation, that is, $X\in \mathcal{G}(n,p_1)$ and $Y\in \mathcal{G}(n,p_2)$. In this paper, we show that if $p_1 p_2 \ge p_0^2=n^{-1+o(1)}$ and $p_1, p_2 \ge w(n) p_0$, where $w(n)\rightarrow 0$ as $n\rightarrow \infty$, then $\mathsf{FS}(X,Y)$ is connected with high probability, which extends the result on $p_1=p_2=p$, due to Alon, Defant, and Kravitz., Comment: 16 pages, 1 fighres. This is a version revised mainly under very careful comments from the anonymous referees. arXiv admin note: substantial text overlap with arXiv:2009.07840 by other authors