Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.Comment: Revised preprint with an extra section on an approximation algorithm. arXiv admin note: text overlap with arXiv:1101.574

What every ICU clinician needs to know about the cardiovascular effects caused by abdominal hypertension

The effects of increased intra-abdominal pressure (IAP) on cardiovascular function are well recognized and include a combined negative effect on preload, afterload and contractility. The aim of this review is to summarize the current knowledge on this topic. The presence of intra-abdominal hypertension (IAH) erroneously increases barometric filling pressures like central venous (CVP) and pulmonary artery occlusion pressure (PAOP) (since these are zeroed against atmospheric pressure). Transmural filling pressures (calculated by subtracting the pleural pressure from the end-expiratory CVP value) may better reflect the true preload status but are difficult to obtain at the bedside. Alternatively, since pleural pressures are seldom measured, transmural CVP can also be estimated by subtracting half of the IAP from the end-expiratory CVP value, since abdominothoracic transmission is on average 50%. Volumetric preload indicators, such as global and right ventricular end-diastolic volumes or the left ventricular end-diastolic area, also correlate better with true preload. When using functional hemodynamic monitoring parameters like stroke volume variation (SVV) or pulse pressure variation (PPV) one must bear in mind that increased IAP will increase these values (via a concomitant increase in intrathoracic pressure). The passive leg raising test may be a false negative in IAH. Calculation of the abdominal perfusion pressure (as mean arterial pressure minus IAP) has been shown to be a better resuscitation endpoint than IAP alone. Finally, it is re-assuring that transpulmonary thermodilution techniques have been validated in the setting of IAH and abdominal compartment syndrome. In conclusion, the clinician must be aware of the different effects of IAH on cardiovascular function in order to assess the volume status accurately and to optimize hemodynamic performance

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675