Universal cycles of classes of restricted words

It is well known that Universal cycles (U-cycles) of k-letter words on an n-letter alphabet exist for all k and n. In this paper, we prove that Universal cycles exist for several restricted classes of words, including non-bijections, “equitable” words (under suitable restrictions), ranked permutations, and “passwords”. In each case, proving the connectedness of the underlying de Bruijn digraph is a non-trivial step.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

最大度大于等于7的平面图的线性荫度

图的染色问题在组合优化、计算机科学和Hessians矩阵的网络计算等方面具有非常重要的应用。其中图的染色中有一种重要的染色——线性荫度,它是一种非正常的边染色,即在简单无向图中,它的边可以分割成线性森林的最小数量。研究最大度$\bigtriangleup（G）\geq7$的平面图$G$的线性荫度,证明了对于两个固定的整数$i$,$j\in\{5,6,7\}$,如果图$G$中不存在相邻的含弦$i$,$j$-圈,则图$G$的线性荫度为$\lceil\frac\bigtriangleup2\rceil$。     

Hamiltonian Cycle Systems Which Are Both Cyclic and Symmetric

The notion of a symmetric Hamiltonian cycle system (HCS) of a graph Γ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for , by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1–15] for , and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs and . In each case, there must be an involutory permutation ψ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for , this ψ should be precisely the permutation switching all pairs of endpoints of the edges of I. An HCS is cyclic if it is invariant under some cyclic permutation of all the vertices. The existence question for a cyclic HCS of has been completely solved by Jordon and Morris [Discrete Math (2008), 2440–2449]—and we note that their cyclic construction is also symmetric for (mod 8). It is then natural to study the existence problem of an HCS of a graph or multigraph Γ as above which is both cyclic and symmetric. In this paper, we completely solve this problem: in the case of even order, the final answer is that cyclicity and symmetry can always cohabit when a cyclic solution exists. On the other hand, imposing that a cyclic HCS of odd order is also symmetric is very restrictive; we prove in fact that an HCS of with both properties exists if and only if is a prime.     

Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

We construct a face two‐colourable, blue and green say, embedding of the complete graph in a nonorientable surface in which there are blue faces each of which have a hamilton cycle as their facial walk and green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of , for all and . Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph , for where and .     

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .     

一类二阶非线性系统的全局渐近稳定性

本文利用微分方程定性理论讨论一类更广泛的二阶非线性系统 dx/dt=h(y)-F(x),dy/dt=-g(x)Q(x,y)零解的全局渐近稳定性.给出了上述系统无环的三个充分条件以及全局渐近稳定的三个定理.     

Cyclical games with prohibitions

We consider a certain combinatorial game on a digraph for two cases of the price function. For one case the game in question extends the cyclical game studied in Ehrenfeucht and Mycielski (1979) and Gurvitch, Karzanov and Khachiyan (1988) which, in its turn, is a generalization of the well-known problem of finding a minimum mean cycle in an edge-weighted digraph. We prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies.This work was partially supported by Chaire municipale, Mairie de Grenoble.     

关于Lins,A.,W.de Melo和Pugh C.C.的猜想(Ⅱ)

在本文中，我们证明了在F(x)＝－F（x）条件下，Lins,A.,W.de Melo和Pugh C．C．的猜想是成立的。