On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.Comment: 39 pages, 12 figures, 6 table

Conduction Properties Distinguish Unmyelinated Sympathetic Efferent Fibers and Unmyelinated Primary Afferent Fibers in the Monkey

Different classes of unmyelinated nerve fibers appear to exhibit distinct conductive properties. We sought a criterion based on conduction properties for distinguishing sympathetic efferents and unmyelinated, primary afferents in peripheral nerves.In anesthetized monkey, centrifugal or centripetal recordings were made from single unmyelinated nerve fibers in the peroneal or sural nerve, and electrical stimuli were applied to either the sciatic nerve or the cutaneous nerve endings, respectively. In centrifugal recordings, electrical stimulation at the sympathetic chain and dorsal root was used to determine the fiber's origin. In centrifugal recordings, sympathetic fibers exhibited absolute speeding of conduction to a single pair of electrical stimuli separated by 50 ms; the second action potential was conducted faster (0.61 0.16%) than the first unconditioned action potential. This was never observed in primary afferents. Following 2 Hz stimulation (3 min), activity-dependent slowing of conduction in the sympathetics (8.6 0.5%) was greater than in one afferent group (6.7 0.5%) but substantially less than in a second afferent group (29.4 1.9%). In centripetal recordings, most mechanically-insensitive fibers also exhibited absolute speeding to twin pulse stimulation. The subset that did not show this absolute speeding was responsive to chemical stimuli (histamine, capsaicin) and likely consists of mechanically-insensitive afferents. During repetitive twin pulse stimulation, mechanosensitive afferents developed speeding, and speeding in sympathetic fibers increased.The presence of absolute speeding provides a criterion by which sympathetic efferents can be differentiated from primary afferents. The differences in conduction properties between sympathetics and afferents likely reflect differential expression of voltage-sensitive ion channels

Large-Scale Screening of a Targeted Enterococcus faecalis Mutant Library Identifies Envelope Fitness Factors

Spread of antibiotic resistance among bacteria responsible for nosocomial and community-acquired infections urges for novel therapeutic or prophylactic targets and for innovative pathogen-specific antibacterial compounds. Major challenges are posed by opportunistic pathogens belonging to the low GC% Gram-positive bacteria. Among those, Enterococcus faecalis is a leading cause of hospital-acquired infections associated with life-threatening issues and increased hospital costs. To better understand the molecular properties of enterococci that may be required for virulence, and that may explain the emergence of these bacteria in nosocomial infections, we performed the first large-scale functional analysis of E. faecalis V583, the first vancomycin-resistant isolate from a human bloodstream infection. E. faecalis V583 is within the high-risk clonal complex 2 group, which comprises mostly isolates derived from hospital infections worldwide. We conducted broad-range screenings of candidate genes likely involved in host adaptation (e.g., colonization and/or virulence). For this purpose, a library was constructed of targeted insertion mutations in 177 genes encoding putative surface or stress-response factors. Individual mutants were subsequently tested for their i) resistance to oxidative stress, ii) antibiotic resistance, iii) resistance to opsonophagocytosis, iv) adherence to the human colon carcinoma Caco-2 epithelial cells and v) virulence in a surrogate insect model. Our results identified a number of factors that are involved in the interaction between enterococci and their host environments. Their predicted functions highlight the importance of cell envelope glycopolymers in E. faecalis host adaptation. This study provides a valuable genetic database for understanding the steps leading E. faecalis to opportunistic virulence

Edge-disjoint Rainbow Spanning Trees in Complete Graphs

Let G role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eG be an edge-colored copy of Kn role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eKn, where each color appears on at most n/2 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en/2 edges (the edge-coloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eG where each edge has a different color. Brualdi and Hollingsworth (1996) conjectured that every properly edge-colored Kn role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eKn (n≥6 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en≥6 and even) using exactly n−1 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en−1 colors has n/2 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en/2 edge-disjoint rainbow spanning trees, and they proved there are at least two edge-disjoint rainbow spanning trees. Kaneko et al. (2003) strengthened the conjecture to include any proper edge-coloring of Kn role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eKn, and they proved there are at least three edge-disjoint rainbow spanning trees. Akbari and Alipour (2007) showed that each Kn role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eKn that is edge-colored such that no color appears more than n/2 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en/2 times contains at least two rainbow spanning trees. We prove that if n≥1,000,000 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en≥1,000,000, then an edge-colored Kn role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eKn, where each color appears on at most n/2 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3en/2 edges, contains at least ⌊n/(1000logn)⌋ role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3e⌊n/(1000logn)⌋ edge-disjoint rainbow spanning trees