We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

BackgroundSpecific dietary and other lifestyle behaviors may affect the success of the straightforward-sounding strategy “eat less and exercise more” for preventing long-term weight gain. MethodsWe performed prospective investigations involving three separate cohorts that included 120,877 U.S. women and men who were free of chronic diseases and not obese at baseline, with follow-up periods from 1986 to 2006, 1991 to 2003, and 1986 to 2006. The relationships between changes in lifestyle factors and weight change were evaluated at 4-year intervals, with multivariable adjustments made for age, baseline body-mass index for each period, and all lifestyle factors simultaneously. Cohort-specific and sex-specific results were similar and were pooled with the use of an inverse-variance–weighted meta-analysis. ResultsWithin each 4-year period, participants gained an average of 3.35 lb (5th to 95th percentile, −4.1 to 12.4). On the basis of increased daily servings of individual dietary components, 4-year weight cha...

/pdf/changes-in-diet-and-lifestyle-and-long-term-weight-gain-in-3s08pxd2ja.pdf

Changes in Diet and Lifestyle and Long-Term Weight Gain in Women and Men

Background Nasal carriers of Staphylococcus aureus are at increased risk for health care–associated infections with this organism. Decolonization of nasal and extranasal sites on hospital admission may reduce this risk. Methods In a randomized, double-blind, placebo-controlled, multicenter trial, we assessed whether rapid identification of S. aureus nasal carriers by means of a real-time polymerase-chain-reaction (PCR) assay, followed by treatment with mupirocin nasal ointment and chlorhexidine soap, reduces the risk of hospital-associated S. aureus infection. Results From October 2005 through June 2007, a total of 6771 patients were screened on admission. A total of 1270 nasal swabs from 1251 patients were positive for S. aureus. We enrolled 917 of these patients in the intention-to-treat analysis, of whom 808 (88.1%) underwent a surgical procedure. All the S. aureus strains identified on PCR assay were susceptible to methicillin and mupirocin. The rate of S. aureus infection was 3.4% (17 of 504 patients) in the mupirocin–chlorhexidine group, as compared with 7.7% (32 of 413 patients) in the placebo group (relative risk of infection, 0.42; 95% confidence interval [CI], 0.23 to 0.75). The effect of mupirocin–chlorhexidine treatment was most pronounced for deep surgical-site infections (relative risk, 0.21; 95% CI, 0.07 to 0.62). There was no significant difference in all-cause in-hospital mortality between the two groups. The time to the onset of nosocomial infection was shorter in the placebo group than in the mupirocin–chlorhexidine group (P = 0.005). Conclusions The number of surgical-site S. aureus infections acquired in the hospital can be reduced by rapid screening and decolonizing of nasal carriers of S. aureus on admission. (Current Controlled Trials number, ISRCTN56186788.)

/pdf/preventing-surgical-site-infections-in-nasal-carriers-of-2scawz93yr.pdf

Preventing Surgical-Site Infections in Nasal Carriers of Staphylococcus aureus

The understanding of nature and nurture within developmental science has evolved with alternating ascendance of one or the other as primary explanations for individual differences in life course trajectories of success or failure. A dialectical perspective emphasizing the interconnectedness of individual and context is suggested to interpret the evolution of developmental science in similar terms to those necessary to explain the development of individual children. A unified theory of development is proposed to integrate personal change, context, regulation, and representational models of development.

/pdf/a-unified-theory-of-development-a-dialectic-integration-of-36m741goon.pdf

A Unified Theory of Development: A Dialectic Integration of Nature and Nurture

Emotion has a substantial influence on the cognitive processes in humans, including perception, attention, learning, memory, reasoning, and problem solving. Emotion has a particularly strong influence on attention, especially modulating the selectivity of attention as well as motivating action and behavior. This attentional and executive control is intimately linked to learning processes, as intrinsically limited attentional capacities are better focused on relevant information. Emotion also facilitates encoding and helps retrieval of information efficiently. However, the effects of emotion on learning and memory are not always univalent, as studies have reported that emotion either enhances or impairs learning and long-term memory (LTM) retention, depending on a range of factors. Recent neuroimaging findings have indicated that the amygdala and prefrontal cortex cooperate with the medial temporal lobe in an integrated manner that affords (i) the amygdala modulating memory consolidation; (ii) the prefrontal cortex mediating memory encoding and formation; and (iii) the hippocampus for successful learning and LTM retention. We also review the nested hierarchies of circular emotional control and cognitive regulation (bottom-up and top-down influences) within the brain to achieve optimal integration of emotional and cognitive processing. This review highlights a basic evolutionary approach to emotion to understand the effects of emotion on learning and memory and the functional roles played by various brain regions and their mutual interactions in relation to emotional processing. We also summarize the current state of knowledge on the impact of emotion on memory and map implications for educational settings. In addition to elucidating the memory-enhancing effects of emotion, neuroimaging findings extend our understanding of emotional influences on learning and memory processes; this knowledge may be useful for the design of effective educational curricula to provide a conducive learning environment for both traditional "live" learning in classrooms and "virtual" learning through online-based educational technologies.

/pdf/the-influences-of-emotion-on-learning-and-memory-4ige7o8mde.pdf

The Influences of Emotion on Learning and Memory.

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. The strong rainbow connection number $src(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices $u$ and $v$ of $G$ are connected by a rainbow path of length $d(u,v)$. In this paper, we give upper and lower bounds of the (strong) rainbow connection Cayley graphs of Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.

The (strong) rainbow connection numbers of Cayley graphs of Abelian groups

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected In this paper, we show that almost all graphs have the conflict-free connection number 2 More precisely, let $G(n,p)$ denote the Erdős-Renyi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independent from other pairs We prove that for sufficiently large $n$, $cfc(G(n,p))\le 2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$ This means that as soon as $G(n,p)$ becomes connected with high probability, $cfc(G(n,p))\le 2$

Conflict-free connection number of random graphs

An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.

Extremal graphs and classification of planar graphs by MC-numbers

Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound.

Xueliang Li

Papers

The (strong) rainbow connection numbers of Cayley graphs of Abelian groups

Conflict-free connection number of random graphs

Extremal graphs and classification of planar graphs by MC-numbers

The Hamiltonian index of a graph and its branch-bonds

Compatible Eulerian circuits in Eulerian (di)graphs with generalized transition systems