Chip firing and all-terminal network reliability bounds

The (all-terminal) reliability of a graph G is the probability that all vertices are in the same connected component, given that vertices are always operational but edges fail independently each with probability p. Computing reliability is #P-complete, and hence is expected to be intractable. Consequently techniques for efficiently (and effectively) bounding reliability have been the major thrust of research in the area. We utilize a deep connection between reliability and chip firings on graphs to improve previous bounds for reliability.     

微机Windows 9X平台上的分子模拟技术

通过例子介绍了微机Windows平台上的分子模拟技术——分子结构的生成、分子力学量子化学计算及计算结果的图形显示等内容。     

Minimal cutwidth linear arrangements of abelian Cayley graphs

We find the minimal cutwidth and bisection width values for abelian Cayley graphs with up to 4 generators and present an algorithm for finding the corresponding optimal ordering. We also find minimal cuts of each order.     

Strong edge-magic graphs of maximum size

An edge-magic total labeling on G is a one-to-one map λ from V(G)∪E(G) onto the integers 1,2,…,|V(G)∪E(G)| with the property that, given any edge (x,y), λ(x)+λ(x,y)+λ(y)=k for some constant k. The labeling is strong if all the smallest labels are assigned to the vertices. Enomoto et al. proved that a graph admitting a strong labeling can have at most 2|V(G)|-3 edges. In this paper we study graphs of this maximum size.     

Path coupling using stopping times and counting independent sets and colorings in hypergraphs

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ + 1. We also show that the Glauber dynamics for proper q‐colorings of a hypergraph mixes rapidly if m ≥ 4 and q > Δ, and if m = 3 and q ≥ 1.65Δ. We give related results on the hardness of exact and approximate counting for both problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

An adaptive memory algorithm for the k-coloring problem

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. The adaptive memory algorithm is a hybrid evolutionary heuristic that uses a central memory. At each iteration, the information contained in the central memory is used for producing an offspring solution which is then possibly improved using a local search algorithm. The so obtained solution is finally used to update the central memory. We describe in this paper an adaptive memory algorithm for the k-coloring problem. Computational experiments give evidence that this new algorithm is competitive with, and simpler and more flexible than, the best known graph coloring algorithms.     

On the arithmetic product of combinatorial species

We introduce two new binary operations on combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series in the context of species. It allows us to introduce the notion of multiplicative species, a lifting to the combinatorial level of the classical notion of multiplicative arithmetic function. Interesting combinatorial constructions are introduced; cloned assemblies of structures, hyper-cloned trees, enriched rectangles, etc. Recent research of Cameron, Gewurz and Merola, about the product action in the context of oligomorphic groups, motivated the introduction of the modified arithmetic product. By using the modified arithmetic product we obtain new enumerative results. We also generalize and simplify some results of Canfield, and Pittel, related to the enumerations of tuples of partitions with the restrictions met.     

A conjecture of Borodin and a coloring of Grünbaum

In 1976, Borodin conjectured that every planar graph has a 5‐coloring such that the union of every k color classes with 1 ≤ k ≤ 4 induces a (k—1)‐degenerate graph. We prove the existence of such a coloring using 18 colors. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:139–147, 2008     

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.