基于PLS的变权重组合预测方法

在目前研究较多的组合预测模型中加权系数是不变的.事实上,假定加权系数为常数,组合预测模型并不能很好地反映预测方法的有效性.基于以上事实,本文提出基于PLS的变权重组合预测方法,利用偏最小二乘回归方法求得组合预测的权重函数.最后通过实例分析验证了方法的有效性.     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ ksum_{i=1}^{d}f_{i}(v)le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{{k}}(G). Note that d_t^{1}(G)d_{t}^{{1}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{{k}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

2-Rainbow domination number of Cartesian products: C_{n}\square C_{3} and C_{n}\square C_{5}

A function \(f:V(G)\rightarrow \mathcal P (\{1,\ldots ,k\})\) is called a \(k\) -rainbow dominating function of \(G\) (for short \(kRDF\) of \(G)\) if \( \bigcup \nolimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},\) for each vertex \( v\in V(G)\) with \(f(v)=\varnothing .\) By \(w(f)\) we mean \(\sum _{v\in V(G)}\left|f(v)\right|\) and we call it the weight of \(f\) in \(G.\) The minimum weight of a \( kRDF\) of \(G\) is called the \(k\) -rainbow domination number of \(G\) and it is denoted by \(\gamma _{rk}(G).\) We investigate the \(2\) -rainbow domination number of Cartesian products of cycles. We give the exact value of the \(2\) -rainbow domination number of \(C_{n}\square C_{3}\) and we give the estimation of this number with respect to \(C_{n}\square C_{5},\) \((n\ge 3).\) Additionally, for \(n=3,4,5,6,\) we show that \(\gamma _{r2}(C_{n}\square C_{5})=2n.\)      

F_{3}-domination problem of graphs

Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\) . A set \(S\subseteq V(G)\) is an \(F_{3}\) -dominating set of \(G\) if every vertex \(v\) is \(F_{3}\) -dominated by a vertex \(w\) in \(S.\) The \(F_{3}\) -domination number of \(G\) , denoted by \(\gamma _{F_{3}}(G)\) , is the minimum cardinality of an \(F_{3}\) -dominating set of \(G\) . In this paper, we study the \(F_{3}\) -domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\) -domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)      

The total {k}-domatic number of wheels and complete graphs

Let k be a positive integer and let G be a graph with vertex set V(G). The total {k}-dominating function (T{k}DF) of a graph G is a function f from V(G) to the set {0,1,2,??,k}, such that for each vertex v??V(G), the sum of the values of all its neighbors assigned by f is at least k. A set {f ₁,f ₂,??,f _d } of pairwise different T{k}DFs of G with the property that $\sum_{i=1}^{d}f_{i}(v)\leq k$ for each v??V(G), is called a total {k}-dominating family (T{k}D family) of G. The total {k}-domatic number of a graph G, denoted by $d_{t}^{\{k\}}(G)$ , is the maximum number of functions in a T{k}D family. In this paper, we determine the exact values of the total {k}-domatic numbers of wheels and complete graphs, which answers an open problem of Sheikholeslami and Volkmann (J. Comb. Optim., 2010) and completes a result in the same paper.     

Weak {2}-domination number of Cartesian products of cycles

For a graph \(G=(V, E)\), a weak \(\{2\}\)-dominating function \(f:V\rightarrow \{0,1,2\}\) has the property that \(\sum _{u\in N(v)}f(u)\ge 2\) for every vertex \(v\in V\) with \(f(v)= 0\), where N(v) is the set of neighbors of v in G. The weight of a weak \(\{2\}\)-dominating function f is the sum \(\sum _{v\in V}f(v)\) and the minimum weight of a weak \(\{2\}\)-dominating function is the weak \(\{2\}\)-domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak \(\{2\}\)-domination number of \(C_n \Box C_5\), which was obtained by St?pień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak \(\{2\}\)-domination numbers of \(C_n \Box C_3\) and \(C_n \Box C_4\).     

On the total {k}-domination number of Cartesian products of graphs

Let γ _t ^{k}(G) denote the total {k}-domination number of graph G, and let denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, . As a corollary of this result, we have for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005). The work was supported by NNSF of China (No. 10701068 and No. 10671191).     

Arcs in $$\mathbb Z^2_{2p}$$

An arc in \(\mathbb Z^2_n\) is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small n.     

Chromatic number of distance graphs generated by the sets {2,3,x,y}

Let D be a set of positive integers. The distance graph generated by D has all integers ? as the vertex set; two vertices are adjacent whenever their absolute difference falls in D. We completely determine the chromatic number for the distance graphs generated by the sets D={2,3,x,y} for all values x and y. The methods we use include the density of sequences with missing differences and the parameter involved in the so called “lonely runner conjecture”. Previous results on this problem include: For x and y being prime numbers, this problem was completely solved by Voigt and Walther (Discrete Appl. Math. 51:197–209, 1994); and other results for special integers of x and y were obtained by Kemnitz and Kolberg (Discrete Math. 191:113–123, 1998) and by Voigt and Walther (Discrete Math. 97:395–397, 1991).