The ratio of the numbers of odd and even cycles in outerplanar graphs

In this paper, we investigate the ratio of the numbers of odd and even cycles in outerplanar graphs. We verify that the ratio generally diverges to infinity as the order of a graph diverges to infinity. We also give sharp estimations of the ratio for several classes of outerplanar graphs, and obtain a constant upper bound of the ratio for some of them. Furthermore, we consider similar problems in graphs with some pairs of forbidden subgraphs/minors, and propose a challenging problem concerning claw-free graphs.     

Global secure sets of grid-like graphs

Let G=(V,E) be a graph and S⊆V. The set S is a secure set if ∀X⊆S,|N[X]∩S|≥|N[X]−S|, and S is a global secure set if S is a secure set and a dominating set. The cardinality of a minimum global secure set of G is the global security number of G, denoted γ_s(G). The sets studied in this paper are different from secure dominating sets studied in Cockayne et al. (2003) [3], Grobler and Mynhardt (2009) [8], or Klostermeyer and Mynhardt (2008) [13], which are also denoted by γ_s.In this paper, we provide results on the global security numbers of paths, cycles and their Cartesian products.     

Elementary submodels in infinite combinatorics

The use of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams’s theorem on cycle decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.     

双阈值二元神经网络极限环的存在惟一性

研究具负反馈的双阈值二元神经网络模型的动力学性质。在一定的初始函数空间内，对设定的阈值范围证明了模型存在惟一稳定极限环。     

实际回热式布雷顿制冷机的性能优化

用有限时间热力学方法分析实际回热式布雷顿制冷机的性能特性,以制冷率和制冷系数为优化目标,优化了高低、温侧换热器和回热器的热导率分配以及工质和热源间的热容率匹配,并采用数值计算分析了各参数值对最优性能的影响特点.所得结果对工程制冷系统设计有一定的指导意义。     

Structural features responsible for GFPuv and S147P-GFP’s improved fluorescence

Green fluorescent protein (GFP) is used as a biological marker. It is a protein in the jellyfish, Aequorea victorea, which is found in the cold Pacific Northwest. Mature GFP, i.e. fully fluorescent GFP, is most efficiently formed at temperatures well below 37 °C. The GFPuv (F99S/M153T/V163A) and S147P-GFP mutants mature more efficiently at room temperature than wild-type GFP, and therefore result in increased fluorescence at room temperature. Computational methods have been used to examine whether the low-energy precyclized forms of these improved GFP-mutants are preorganized so that they can more efficiently form the chromophore than the wild-type and S65T-GFP. All mutations examined (S147P, F99S, M153T, V163A and F99S/M153T/V163A) more efficiently preorganize the immature precyclized forms of GFP for chromophore formation than immature wild-type GFP. It has been proposed that Arg96 is involved in chromophore formation. Our calculations suggest that the M153T and V163A mutations in GFPuv maybe partially responsible for the increased maturation efficiency observed in GFPuv because they improve the Arg96–Tyr66 interaction. The same is true for the S147P mutation in S147P-GFP.     

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

Strong embeddings of minimum genus

A “folklore conjecture, probably due to Tutte” (as described in [P.D. Seymour, Sums of circuits, in: Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, 1977), Academic Press, 1979, pp. 341-355]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. Sporadic counterexamples to this conjecture have been known since the late 1970s. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing a plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.     

Cycle factors in strongly connected local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A digraph is 2-connected if the removal of an arbitrary vertex results in a strongly connected digraph.In 2004 and 2005, Li and Shu investigated the structure of strongly connected, but not 2-connected tournaments. Using their structural results they were able to give sufficient conditions for a strongly connected tournament T to have complementary cycles or a k-cycle factor, i.e. a set of k vertex disjoint cycles that span the vertex set of T.Inspired by the articles of Li and Shu we develop in this paper the structure necessary for a strongly connected local tournament to be not cycle complementary. Using this structure, we are able to generalize and transfer various results of Li and Shu to the class of local tournaments.     

Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, and very recently the conjecture was proved for the case where r=3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for tournaments with minimum in-degree at least 2r−1. In particular, this shows that the conjecture is true for (almost) regular tournaments. In the first proof, we prove auxiliary results about union of sets contained in another union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.