Rainbow H-factors of complete s-uniform r-partite hypergraphs

We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V-1, V-2, ..., V-r} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k vertices in each vertex class by T-s,T-r(k). In this paper we prove that if h, r and s are positive integers with 2 <= s <= r <= h then there exists a constant k = k(h, r, s) so that if H is an s-uniform hypergraph with h vertices and chromatic number X(H) = r then any proper edge coloring of T-s,T-r(k) has a rainbow H-factor

Clar and Sextet Polynomials of Boron2nitrogen Fullerenes

摘要: Clar 结构因其在比较分子稳定性中的作用而广受关注. Shiu W C 等人计算了C60的Clar 结构的数量并给出其Clar 多项式和sextet 多项式,而对于硼氮富勒烯,相应的问题尚未解决. 本文考查了Seifert G等人确定的最稳定的3 种硼氮富 勒烯的结构特征,通过组合原理得到B12 N12 ,B16 N16 的Clar 多项式和Sextet 多项式,并给出了详细证明. 此外还给出用于 一般硼氮富勒烯的计算程序,并作为例子给出B28N28的Clar 多项式和Sextet 多项式. 本文的结果解决了一般的硼氮富勒 烯分子Clar 多项式和Sextet 多项式的计算工作. Abstract : Shiu W C et al. computed the count of Clar st ructures of C60 and the associated Clar polynomial and sextet polynomial. For boron2nit rogen fullerenes ,the corresponding problem has not been solved. This paper considers three BN2fullerenes which were a2 nomalously stable. By combinatorial techniques the Clar polynomial and the sextet polynomial of B12N12 and B16N16 are given. Detailed proof is also provided. Furthermore ,a program for general cases is designed. As an illust ration ,the Clar polynomial and the sextet pol2 ynomial of B28N28 is given.国家自然科学基金(10371102) 资

循环图的支撑树数与Euler环游数的渐近计数定理

研究有向循环图C( P ,S1,S2 ,… ,Sk)支撑树数与EulEr环游数的渐近性质 ,得到其支撑树数T(C( P ,S1,S2 ,… ,Sk) )与EulEr环游数E(C( P ,S1,S2 ,… ,Sk) )的渐近公式lIM 1kP T(C( P ,S1,S2 ,… ,Sk) ) =1 ,lIM 1k !P E(C( P ,S1,S2 ,… ,Sk) ) =1 ,　　　　P→∞ .在此基础上得到了其叠线图EulEr环游与支撑树数的渐近公式 .对无向图也得到了平行的结果 .国家自然科学基金!(批准号:69673042);香港CERG基金资助项

循环图的支撑树数与Euler 环游数的渐近计数定理

摘要　　研究有向循环图C( p , s1 , s2 , ⋯, sk ) 支撑树数与Euler 环游数的渐近性质, 得到其支撑树数T ( C( p , s1 , s2 , ⋯, sk) ) 与Euler 环游数E( C( p , s1 , s2 , ⋯, sk) ) 的渐 近公式 lim 1 k p T ( C( p , s1 , s2 , ⋯, sk) ) = 1 , lim 1 k ! p E( C( p , s1 , s2 , ⋯, sk) ) = 1 , 　　　　p → ∞. 在此基础上得到了其叠线图Euler 环游与支撑树数的渐近公式. 对无向图也得到了 平行的结果

Limit points of eigenvalues of (di)graphs

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D, the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set D of digraphs (graphs) is a limit point of eigenvalues of a set of bipartite digraphs (graphs), where consists of the double covers of the members in D. 4. Every limit point of eigenvalues of a set D of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in D. 5. If M is a limit point of the largest eigenvalues of graphs, then -M is a limit point of the smallest eigenvalues of graphs

Perfect matchings of polyomino graphs

This paper gives necessary and sufficient conditions for a polyomino graph to have a perfect matching and to be elementary, respectively. As an application, we can decompose a non-elementary polyomino with perfect matchings into a number of elementary subpolyominoes so that the number of perfect matchings of the original non-elementary polyomino is equal to the product of those of the elementary subpolyominoes

Interpolation theorem for a continuous function on orientations of a simple graph

Let G be a simple graph. A function f from the set of orientations of G to the set of Iron-negative integers is called a continuous function on orientations of G if, for any two orientations O-1 and O-2 of G, \f(O-1) - f(O-2)\ less than or equal to 1 whenever O-1 and O-2 differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O-1 and O-2 of G with f(O-1) = p and f(O-2) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G

若干与统计物理相关的组合问题

简单介绍统计物理与纽结理论中的一些组合问题,包括反射对称图的dimer问题与支撑树的计数问题,Dimer问题的熵与边界的关系,计数平面图的完美匹配数的图压缩方法,链环多项式的计算以及Jones多项式的零点的分布问题,主要综述近年来我们在这些问题中得到的部分结果

On the location of zeros of the Homfly polynomial

National Natural Science Foundation of China [10831001]; Fundamental Research Funds for the Central Universities [2010121007]Wu, Wang, Chang and Shrock initiated the study of zeros of the Jones polynomial since it was the special case of partition functions of the Potts model in physics. The Homfly polynomial is the generalization of the Jones polynomial. Let L be an oriented link, and P(L)(v, z) be its Homfly polynomial. In this paper, we study zeros of P(L)(v, z) with z fixed. We prove the so-called unit-circle theorem for a family of generalized Jaeger's links {Dn(G)| n = 1, 2, ...} which states that |v| = 1 is the limit of zeros of Homfly polynomials of generalized Jaeger's links {Dn(G)| n = 1, 2, ...} if G is bridgeless. Similar to the result of the Jones polynomial, we also prove that zeros of Homfly polynomials are dense in the whole complex plane

Enumerating tree-like polyphenyl isomers

NSFC [10831001]Enumeration of molecules is one of the fundamental problems in bioinformatics and plays an important role in drug discovery, experimental structure elucidation (e.g., by using NMR or mass spectrometry), molecular design and virtual library construction. We consider the enumeration of tree-like polyphenyls (C(6)nH(4n+2)). For this purpose, we de fine two generating functions T (x) and R (x) involving the numbers t(n) and r(n) of tree-like polyphenyls (TL-polyphenyls) and monosubstituted tree-like polyphenyls (MTL-polyphenyls), respectively. By characterizing the symmetry groups with respect to TL-polyphenyls and MTL-polyphenyls, we establish two functional equations for these two generating functions. This yields for the first time an efficient recursion formula for calculating the numbers t(n) and r(n). The two functional equations are also the fundamentals for analyzing their asymptotic behaviors, from which we derive the precise asymptotic values for both r(n) and t(n). The resulting asymptotic values are shown to fit well to the numerical results obtained by using our recursion formula. Finally, we give an explicit enumerating expression for TL-polyphenyls of a particular type: the linear polyphenyls