On Locally Primitive Graphs of Order 18p

The authors investigate locally primitive bipartite regular connected graphs of order $18p$. It is shown that such a graph is either arc-transitive or isomorphic to one of the Gray graph and the Tutte $12$-cage.     

Strong Circuit Double Cover of Some Cubic Graphs

Let C be a given circuit of a bridgeless cubic graph G. It was conjectured by Seymour that G has a circuit double cover (CDC) containing the given circuit C. This conjecture (strong CDC [SCDC] conjecture) has been verified by Fleischner and Häggkvist for various families of graphs and circuits. In this article, some of these earlier results have been improved: (1) if contains a Hamilton path or a Y‐tree of order less than 14, then G has a CDC containing C; (2) if is connected and , then G has a CDC containing C.     

A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

Implications among linkage properties in graphs

Given a graph H with vertices w₁, …, w_m, a graph G with at least m vertices is H‐linked if for every choice of vertices v₁, …, v_m in G, there is a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i ). This concept generalizes the notions of k‐linked, k‐connected, and k‐ordered graphs. For graphs H₁ and H₂ with the same order that are not contained in stars, the property of being H₁‐linked implies that of being H₂‐linked if and only if H₂?H₁. The implication also holds when H₁ is obtained from H₂ by replacing an edge xy with an edge from y to a new vertex x′. Other instances of nonimplication are obtained, using a lemma that the number of vertices appearing in minimum vertex covers of a graph G is at most the vertex cover number plus the size of a maximum matching. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 327‐337, 2009     

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C₁,…,C_k} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.     

直径为5的图的2-导出匹配划分和2-导出匹配覆盖问题的NP-完全性

给定一个简单图G和正整数κ,具有完美匹配的图G的κ-导出匹配划分是对顶点集V(C)的一个κ-划分(V1,V2,...,Vκ),其中对每一个i(1≤i≤κ),由Vi导出的G的子图G[Vi]是1-正则的.κ-导出匹配划分问题是指对给定的图G,判定G是否存在一个κ-导出匹配划分.令M1,M2…,Mκ为图G的κ个导出匹配,如果V(M1)UV(M2)∪...∪V(Mκ)=V(G),则我们称{M1,M2,...,Mκ}是G的κ-导出匹配覆盖.κ-导出匹配覆盖问题是指对给定的图G,判定G是否存在κ-导出匹配覆盖.本文给出了Yang,Yuan和Dong所提出问题的解,证明了直径为5的图的导出匹配2一划分问题和导出匹配2-覆盖问题都是NP-完全的.     

图的局部减边控制数

引入局部减边控制函数和局部减边控制数的概念,得到了图的最小局部减边控制函数的性质,给出了局部减边控制数的最好上下界,确定了一些特殊图的局部减边控制数.最后得到了图的减边控制数的最好上界.     

Representing the sporadic Archimedean polyhedra as abstract polytopes

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterize which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).     

精确覆盖问题的加权分治算法

精确覆盖问题是组合优化中经典的NP-Hard问题之一,其在诸多领域具有广泛的应用价值。本文首先研究了精确覆盖问题的数学性质,并根据数学性质提出相应的分支降阶规则以缩小问题的规模;接着设计了一个基于分支降阶的回溯算法求解该问题;然后运用常规技术分析得出该精确算法的时间复杂度为O(1.4656^k);最后运用加权分治技术对该算法的时间复杂度进行分析,将该算法的时间复杂度降为O(1.3842^k)。文章最后通过一个示例进一步阐述该算法的原理,并与其他精确算法进行了对比分析,研究结果表明该算法是可行的,也是有效的。     

Operations on 3-Dimensional Small Covers

The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura＇s theorem on the construction of oriented 3-dimensional small covers and the Lu-Yu＇s theorem on the construction of all 3-dimensional small covers are improved. Moreover, for a construction of 3-dimensional 2-torus manifolds, it is shown that all operations can be obtained by using the equivariant surgeries.