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Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ j,k -number of G, denoted by λ j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| m ≥j if u and v are adjacent; and |f(u)−f(v)| m ≥k if u and v are at distance two apart, where |x| m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ j,k -number of G and denoted by σ j,k (G). The λ j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ j,k -numbers of direct products of two complete graphs and the σ j,k -numbers of direct products and Cartesian products of two complete graphs.

Green stocks are companies environmental protective and friendly. We test Green stock index in Shanghai Stock Exchange and China Securities Index as safe-havens for global investors. Suitable multivariate-SV model and Bayesian method are used to estimate the spillover effect between different assets among local and global markets. We choose multivariate volatility model because it can efficiently simulate the spillover effect by using machine learning MCMC method. The results show that the Environmental Protection Index (EPI) of Shanghai Stock Exchange (SSE) and China Securities Index (CSI) have no significant volatility spillover from Shanghai Stock index, S &P index, gold price, oil future prices of USA and China. During COVID-19 pandemic, we find Green stock index is a suitable safe-haven with low volatility spillover. Green stock indexes has a strongly one-way spillover to the crude oil future price. Environmentally friendly investor can use diversity green assets to provide a low risk investment portfolio in EPI stock market. The DCGCt-MSV model using machine learning of MCMC method is accurate and outperform others in Bayes parameter estimation.

A vertex coloring is called $$2$$ -distance if any two vertices at distance at most $$2$$ from each other get different colors. The minimum number of colors in 2-distance colorings of $$G$$ is its 2-distance chromatic number, denoted by $$\chi _2(G)$$ . Let $$G$$ be a plane graph with girth at least $$5$$ . In this paper, we prove that $$\chi _2(G)\le \Delta +8$$ for arbitrary $$\Delta (G)$$ , which partially improves some known results.

Inspired by the applications in on-demand manufacturing, we introduce the online k-color spanning disk problem, the first online model for color spanning problems to the best of our knowledge. Given a set P of n colored points in a plane, with each color chosen from a set C of $$m \le n$$ colors, the online k-color spanning disk problem determines the location of the center that minimizes the accumulated radius of the minimum spanning disks for a sequence of color sets, denoted by $$\delta =\langle C_1,C_2,\ldots ,C_T\rangle $$ , $$C_t\subseteq C$$ , $$|C_t| \ge k$$ , $$t\in \{1, 2, \ldots , T\}$$ , as they are presented online. Here, a minimum spanning disk for a color set means a disk contains at least one point of each color. We construct a special instance to establish a lower bound on the performance of any online algorithms. Then, an $$O(nm\log n)$$ -time Voronoi-diagram-based algorithm is designed such that its competitive ratio matches the problem’s lower bound. This implies our algorithm is theoretically the best possible in terms of the competitive ratio. We also introduce and study a variant, named the online balanced k-color spanning disk problem, for which a non-trivial lower bound and a best possible algorithm are presented.

A fully odd K4 is a subdivision of K4 such that each of the six edges of the K4 is subdivided into a path of odd length. In 1974, Toft conjectured that every graph containing no fully odd K4 can be vertex-colored with three colors. The purpose of this paper is to prove Toft's conjecture.

Improper value of the parameter p in robust constraints will result in no feasible solutions while applying stochastic p-robustness optimization approach (p-SRO) to solving facility location problems under uncertainty. Aiming at finding the lowest critical p-value of parameter p and corresponding robust optimal solution, we developed a novel robust optimization approach named as min-p robust optimization approach (min-pRO) for P-median problem (PMP) and fixed cost P-median problem (FPMP). Combined with the nearest allocation strategy, the vertex substitution heuristic algorithm is improved and the influencing factors of the lowest critical p-value are analyzed. The effectiveness and performance of the proposed approach are verified by numerical examples. The results show that the fluctuation range of data is positively correlated with the lowest critical p-value with given number of new facilities. However, the number of new facilities has a different impact on lowest critical p-value with the given fluctuation range of data. As the number of new facilities increases, the lowest critical p-value for PMP and FPMP increases and decreases, respectively.

Product family concept predicates to a products’ group or services that derive from a common base and they have usually the same production process and physical features, in different industries. In this paper, we assume two manufacturers that produce some homogeneous product families that they want to decide upon their wholesale price and national advertising expenditure, as a new problem in the multi-agent environment. According to this problem, we propose the non-cooperative, cooperative and two-stage game models to maximize manufacturers' profits and then, based on the optimization approaches i.e., Nash and Stackelberg methods, we obtain the appropriate equilibrium strategies in a special example. Also, we propose a new profit-sharing approach for the Stackelberg model of the defined problem and discuss the solutions as some propositions and marginal points, based on the numerical studies. The results show that the centralized model leads to better profit than the non-centralized and Stackelberg models.