Spanning Eulerian subgraphs and Catlin’s reduced graphs

A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following: (i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large; (ii) the size of a maximum matching in G is at most 6; (iii) the independence number of G is at most 5. These are improvements of prior results in [16], [18], [24] and [25]

Scaling in the distribution of intertrade durations of Chinese stocks

The distribution of intertrade durations, defined as the waiting times between two consecutive transactions, is investigated based upon the limit order book data of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. A scaling pattern is observed in the distributions of intertrade durations, where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve. The scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions. The ensemble distributions for all stocks are modeled by the Weibull and the Tsallis $q$-exponential distributions. Maximum likelihood estimation shows that the Weibull distribution outperforms the $q$-exponential for not-too-large intertrade durations which account for more than 98.5% of the data. Alternatively, nonlinear least-squares estimation selects the $q$-exponential as a better model, in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities. The distribution of intertrade durations is Weibull followed by a power-law tail with an asymptotic tail exponent close to 3.Comment: 16 elsart pages including 3 eps figure

Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

High Chern number quantum anomalous Hall phases in graphene ribbons with Haldane orbital coupling

We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in a zigzag graphene ribbon under the influence of the exchange field. The effective tight-binding Hamiltonian for graphene is made up of the hopping term, the Kane-Mele and Rashba spin-orbit couplings as well as the Haldane orbital term. We find that the variation of the exchange field results in bulk gap-closing phenomena and phase transitions occur in the graphene system. If the Haldane orbital coupling is absent, the phase transition between the chiral (anti-chiral) edge state $\nu=+2$ ($\nu=-2$) and the pseudo-quantum spin Hall state ($\nu=0$) takes place. Surprisingly, when the Haldane orbital coupling is taken into account, an intermediate QSH phase with two additional edge modes appears in between phases $\nu=+2$ and $\nu=-2$. This intermediate phase is therefore either the hyper-chiral edge state of high Chern number $\nu=+4$ or anti-hyper-chiral edge state of $\nu=-4$ when the direction of exchange field is reversed. We present the band structures, edge state wave functions and current distributions of the different QAH phases in the system. We also report the critical exchange field values for the QAH phase transitions.Comment: 4 figure