边赋权图支撑树权的介值性

有关图的介值性,已有若干较好的结果,但这些结果都未涉及图的赋权。本文考虑边赋权图支撑树权的介值性。证明了定理A 设双射w:E(G)→{1,2,…|E(G)|}是连通图G=(V(G),E(G))的边赋权,如果存在G的圈基使w在它的每个圈上的象集为连续整数集,则w((G))={w(e)|T∈J(G)}是连续整数集。其中(G)是G的支撑树的集合。     

二分图中度条件和k-因子的存在性

本文主要研究了二分图中任意一对距离为２的顶点的度数与ｋ－因子关系,给出了二分图有ｋ因子的若干充分条件,并说明这些条件是最好的可能,从而证明了Ｎｉｓｈｉｍｕｒａ提出的问题对二分图成立。     

关于图划分问题的改进的近似算法

本文考虑NP-难的极大图划分(MAX-GP)问题.我们给出应用半定规划(SDP) 松弛的一个一般方法,并且给出包括极大方向割,稠密子图,极大顶点覆盖,极大割,和极大反割在内的图划分问题的改进的近似比.     

平衡二部图的四圈覆盖

对任意的正整数ｋ,如果每部２ｋ个点的平衡二部图Ｇ＝（ｖ１,ｖ２;Ｅ）的最小度大于等于４ｋ／３,那么Ｇ恰好被ｋ个相互独立的四圈覆盖。     

点染色3-边划分图的最小度(英文)

Addrio-Berry L[1]已经证明了最小度至少为1000的图可以点染色3-边划分,在本文中,我们将其结果改进到了最小度至少为662.     

积图Pm×C4n的k-优美性

Let k be a positive integer. The simple graph G=(V, E) is called k-graceful if there exists a injection.     

图的导出森林2-划分

对一个给定的简单图Ｇ，是否存在Ｖ（Ｇ）的一个２－划分（Ｖ１，Ｖ２）使得每个导出子图Ｇ［Ｖｉ］为森林？称该问题为导出森林２－划分问题．本文证明了对最大度为５的图该问题是ＮＰ－完全的，而对最大度≤４的图该问题多项式时间可解．     

k-方图的邻点可区别无圈边染色

图G的一个正常边染色被称作邻点可区别无圈边染色,如果G中无二色圈,且相邻点关联边的色集合不同.图G的邻点可区别无圈边色数记为χ′_^(aa)(G),即图G的一个邻点可区别无圈边染色所用的最少颜色数.通过构造具体染色的方法,给出了一些k-方图的邻点可区别无圈边色数.     

谱半径前六位的n阶单圈图

恰含一个圈的简单连通图称为单圈图。Cn记n个顶点的圈。△（i,j,κ）记C3的三个顶点上分别接出i,j,κ条悬挂边所得的图，其中i≥j≥κ≥0.Sl^n-l记Cl的某一顶点上接出n-l条悬挂边所得到的图。△（n-4 1,0,0)记△（n-4,0,0)的某个悬挂点上接出一条悬挂边所得到的图。本文证明了：若把所有n(n≥12）阶单圈图按其最大特征值从大到小的顺序排列，则排在前六位的依次是S3^n-3，△（n-4,1,0),△（n-4 1,0,0),S4^n-4,△（n-5,2,0),△（n-5,1,1)。     

On the Efficiency of a Global Non-differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning

The examined algorithm for global optimization of the multiextremal non-differentiable function is based on the following idea: the problem of determination of the global minimum point of the function f(x) on the set (f(x) has a finite number of local minima in this domain) is reduced to the problem of finding all local minima and their attraction spheres with a consequent choice of the global minimum point among them. This reduction is made by application of the optimal set partitioning method. The proposed algorithm is evaluated on a set of well-known one-dimensional, two-dimensional and three-dimensional test functions. Recommendations for choosing the algorithm parameters are given.     

An Adaptive Algorithm for Vector Partitioning

The vector partition problem concerns the partitioning of a set A of n vectors in d-space into p parts so as to maximize an objective function c which is convex on the sum of vectors in each part. Here all parameters d, p, n are considered variables. In this paper, we study the adjacency of vertices in the associated partition polytopes. Using our adjacency characterization for these polytopes, we are able to develop an adaptive algorithm for the vector partition problem that runs in time O(q(L)v) and in space O(L), where q is a polynomial function, L is the input size and v is the number of vertices of the associated partition polytope. It is based on an output-sensitive algorithm for enumerating all vertices of the partition polytope. Our adjacency characterization also implies a polynomial upper bound on the combinatorial diameter of partition polytopes. We also establish a partition polytope analogue of the lower bound theorem, indicating that the output-sensitive enumeration algorithm can be far superior to previously known algorithms that run in time polynomial in the size of the worst-case output.     

Partitioning a Graph into Highly Connected Subgraphs

Given , a k‐proper partition of a graph G is a partition of such that each part P of induces a k‐connected subgraph of G. We prove that if G is a graph of order n such that , then G has a 2‐proper partition with at most parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that if G is a graph of order n with minimum degree where , then G has a k‐proper partition into at most parts. This improves a result of Ferrara et al. ( Discrete Math 313 (2013), 760–764), and both the degree condition and the number of parts is best possible up to the constant c.     

Lock-Gain Based Graph Partitioning

We propose a new heuristic for the graph partitioning problem. Based on the traditional iterative improvement framework, the heuristic uses a new type of gain in selecting vertices to move between partitions. The new type of gain provides a good explanation for the performance difference of tie-breaking strategies in KL-based iterative improvement graph partitioning algorithms. The new heuristic performed excellently. Theoretical arguments supporting its efficacy are also provided. As the proposed heuristic is considered a good candidate for local optimization engines in metaheuristics, we combined it with a genetic algorithm as a sample case and obtained a surprising result that even the average results over 1,000 runs equalled the best known for most graphs.     

Partitioning a Graph into Global Powerful k-Alliances

A set S of vertices of a graph is a defensive k-alliance if every vertex ${v\in S}$ has at least k more neighbors in S than it has outside of S. Analogously, a set S is an offensive k-alliance if every vertex in the neighborhood of S has at least k more neighbors in S than it has outside of S. Also, a powerful k-alliance is a set S of vertices of the graph, which is both defensive k-alliance and offensive (k?+?2)-alliance. A powerful k-alliance is called global if it is a dominating set. In this paper we show that for k?≥ 0, no graph is partitionable into global powerful k-alliances and, for k?≤ ?1, we obtain upper bounds on the maximum number of sets belonging to a partition of a graph into global powerful k-alliances. In addition, we study the close relationships that exist between partitions of a Cartesian product graph, Γ₁?× Γ₂, into (global) powerful (k ₁?+?k ₂)-alliances and partitions of Γ _i into (global) powerful k _i -alliances, ${i\in \{1,2\}}$ .     

Partitioning Boolean lattices into antichains

Let f(n) be the smallest integer t such that a poset obtained from a Boolean lattice with n atoms by deleting both the largest and the smallest elements can be partitioned into t antichains of the same size except for possibly one antichain of a smaller size. In this paper, it is shown that f(n)b n²/log n. This is an improvement of the best previously known upper bound for f(n).     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

太阳图与路的笛卡儿积图的任意可分性（英文）

一个阶为n的图G称为是任意可分的(简作AP),如果对于任一正整数序列τ=(n₁,n₂,…,n_k)满足n=n₁+n₂+…+n_k,总是存在顶点集V(G)的一个划分(V₁,V₂,…,V_k)满足:对于i∈[1,k],|V_i|=n_i,且子图G|V_i|是图G的V_i导出的一个连通子图.我们用S~*=S(n;m₁,m₂,…,m_n)来表示最大度△(S~*)=3的太阳图.本文讨论了图S~*P_m(m≥3)的任意可分性.