Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of a graph $G$ is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

Achieving the Optimal Steaming Capacity and Delay Using Random Regular Digraphs in P2P Networks

In earlier work, we showed that it is possible to achieve $O(\log N)$ streaming delay with high probability in a peer-to-peer network, where each peer has as little as four neighbors, while achieving any arbitrary fraction of the maximum possible streaming rate. However, the constant in the $O(log N)$ delay term becomes rather large as we get closer to the maximum streaming rate. In this paper, we design an alternative pairing and chunk dissemination algorithm that allows us to transmit at the maximum streaming rate while ensuring that all, but a negligible fraction of the peers, receive the data stream with $O(\log N)$ delay with high probability. The result is established by examining the properties of graph formed by the union of two or more random 1-regular digraphs, i.e., directed graphs in which each node has an incoming and an outgoing node degree both equal to one

Light Hadron Spectrum in Quenched Lattice QCD with Staggered Quarks

Without chiral extrapolation, we achieved a realistic nucleon to (\rho)-meson mass ratio of (m_N/m_\rho = 1.23 \pm 0.04 ({\rm statistical}) \pm 0.02 ({\rm systematic})) in our quenched lattice QCD numerical calculation with staggered quarks. The systematic error is mostly from finite-volume effect and the finite-spacing effect is negligible. The flavor symmetry breaking in the pion and (\rho) meson is no longer visible. The lattice cutoff is set at 3.63 (\pm) 0.06 GeV, the spatial lattice volume is (2.59 (\pm) 0.05 fm)(^3), and bare quarks mass as low as 4.5 MeV are used. Possible quenched chiral effects in hadron mass are discussed.Comment: 5 pages and 5 figures, use revtex

Monte Carlo Study of the S=1/2 and S=1 Heisenberg Antiferromagnet on a Spatially Anisotropic Square Lattice

We present a quantum Monte Carlo study of a Heisenberg antiferromagnet on a spatially anisotropic square lattice, where the coupling strength in the x-direction ($J_x$) is different from that in the y-direction ($J_y$). By varying the anisotropy $\alpha$ from 0 to 1, we interpolate between the one-dimensional chain and the two-dimensional isotropic square lattice. Both $S=1/2$ and S=1 systems are considered separately in order to facilitate comparison. The temperature dependence of the uniform susceptibility and the spin-spin correlation length are computed down to very low temperatures for various values of $\alpha$. For S=1, the existence of a quantum critical point at $\alpha^{S=1}_c=0.040(5)$ as well as the scaling of the spin gap is confirmed. Universal quantities predicted from the ${\cal O}(3)$ nonlinear $\sigma$ model agree with our results at $\alpha=0.04$ without any adjustable parameters. On the other hand, the $S=1/2$ results are consistent with $\alpha^{S=1/2}_c=0$, as discussed by a number of previous theoretical studies. Experimental implications for $S=1/2$ compounds such as Sr$_2$CuO$_3$ are also discussed.Comment: 8 pages, 7 figures, to be published in Phys. Rev.