Cubic graphs with large circumference deficit

The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544]

Graphs with equal girth and circumference

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.Comment: 15 pages, comments welcom

The Effect of Dietary Energy and Protein Levels on Production in Breeding Female Ostriches.

1. In a study spanning two breeding seasons, we assessed the effect of different dietary energy and protein levels on body mass, body condition, and egg production of female ostriches. 2. During the first breeding season, groups were given diets with energy concentrations of 8.5, 9.5 and 10.5 MJ/kg dry mass (DM) metabolisable energy (ME) and protein concentrations of 135, 150 and 165 g/kg. In the second breeding season, groups were given diets with ME of 7.5, 8.5 and 9.5 MJ/kg and protein contents of 105, 120 and 135 g/kg. 3. Body mass of birds on diets of 7.5 and 8.5 MJ/kg ME decreased significantly in the course of the breeding season compared with birds fed on diets with higher energy contents and body measurements decreased, suggesting a loss of body condition. 4. Females fed on diets containing only 7.5 MJ/kg ME produced significantly fewer eggs at significantly longer intervals, resulting in fewer chicks hatched. 5. There was no significant difference in egg mass, initial chick mass, chick survival to one month of age and body mass of chicks at one month. 6. Dietary protein concentrations had no effect on egg production, egg mass, hatchability, initial chick mass, chick survival or chick mass at one month old. 7. The female ostriches regained their original body mass during the 4-month rest period between breeding seasons, but significant differences in some parameters during the second breeding season suggest that they may not have fully recovered their body condition. 8. A dietary energy content of 7.5 MJ/kg proved to have an adverse effect on egg production by breeding female ostriches, and it may be concluded from this study that a diet containing 8.5MJ ME/kg DM and 105 g/kg protein should be regarded as the minimum that can be used for breeding female ostriches without compromising egg production

The variation in pressures exerted by commercially available compression garments

Commercially available compression garments (CGs) demonstrate the enhanced recovery from exercise in some, but not all studies. It is possible that in some cases the degree of compression pressure (ComP) exerted is not sufficient to produce any physiological benefit. The aim of this investigation was to identify the levels of ComP exerted by commercially available CGs. This study was composed of two parts. In part A 50 healthy, physically active individuals (n=26 male, n=24 female) were fitted with CGs according to manufacturer’s guidelines. ComP was measured in participants standing in the anatomical position with a pressure measurement device inserted between the skin and the garment. Data were compared to ‘ideal’ pressure values proposed in the literature. In part B ComP in three different brands of CG were compared in a population of 29 men who all wore a medium sized garment. A one way ANOVA indicated that there was a significant difference (P0.05) between observed and ideal pressures in the calf of the male population. No significant differences in pressure (P>0.05) were observed between CG brands at the quadriceps or calf. In conclusion a large number of individuals may not be experiencing an adequate ComP from CG, and this is true for all 3 of the major brands of CGs tested in this investigation

New Bounds for the Dichromatic Number of a Digraph

The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a $k$-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise non-adjacent vertices). The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. Given a digraph $D$, we will construct three different graphs whose chromatic numbers bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$ and $p$, with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length $r$ modulo $\lceil \frac{k}{p} \rceil p$ for each $r\in \{-p+2,\ldots,0,\ldots,p\}$, then $\chi_A (D)\leq \lceil \frac{k}{p} \rceil$; iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves, substantially, the bound proposed by Bondy. Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now.Comment: 14 page

Brief, Aerobic-surge Exercises for Effective Weight Loss: a Randomized, Controlled Trial

Background: Obesity is a growing disease and a consistently effective protocol is needed to reduce this epidemic. The purpose of this study was to determine if a frequent, brief (2-minute) high intensity aerobic exercise (≥75% max HR) was effective in reducing subjects’ weight (BMI) and girth sizes. Methods: A randomized, controlled trial lasting 60 days was conducted at three sites and forty-six subjects completed this study (mean age 39 ± 8 and BMI 32 ±2). The Experimental group was shown how to make movements such as riding a stationary bicycle or lifting dumbbells into an ‘Aerobic-surge’ exercise at or above 75% of calculated maximum heart rate. Control subjects were simply told to “exercise more.” No dietary changes were made for either group. The dependent variables were changes in weight and ‘body summation’ of 10 girth measurements. A 2x2 ANOVA was used to calculate differences. A post-hoc analysis of changes in BMI was also calculated. Results: There was a significant difference between groups (P<0.0001). The mean change of the groups were as follows: Experimental (N=23) mean reductions = -18.lbs and -18.7” vs. Control (N=23) mean changes = +1.3.lbs and +1.3.” Conclusion: The repeated, brief (2-minute) aerobic-surge exercise protocol, performed 4.2 times/day was effective in reducing subjects’ weight and body circumferences. Isolated tests of body composition showed that change were primarily in body fat. This could be a tool in reducing the obesity epidemic. The Aerobic-surge exercise was effective without a dietary changes. Retrospectively Registered Trial: ISRCTN 17326333

Impact of facial conformation on canine health: Brachycephalic Obstructive Airway Syndrome

The domestic dog may be the most morphologically diverse terrestrial mammalian species known to man; pedigree dogs are artificially selected for extreme aesthetics dictated by formal Breed Standards, and breed-related disorders linked to conformation are ubiquitous and diverse. Brachycephaly–foreshortening of the facial skeleton–is a discrete mutation that has been selected for in many popular dog breeds e.g. the Bulldog, Pug, and French Bulldog. A chronic, debilitating respiratory syndrome, whereby soft tissue blocks the airways, predominantly affects dogs with this conformation, and thus is labelled Brachycephalic Obstructive Airway Syndrome (BOAS). Despite the name of the syndrome, scientific evidence quantitatively linking brachycephaly with BOAS is lacking, but it could aid efforts to select for healthier conformations. Here we show, in (1) an exploratory study of 700 dogs of diverse breeds and conformations, and (2) a confirmatory study of 154 brachycephalic dogs, that BOAS risk increases sharply in a non-linear manner as relative muzzle length shortens. BOAS only occurred in dogs whose muzzles comprised less than half their cranial lengths. Thicker neck girths also increased BOAS risk in both populations: a risk factor for human sleep apnoea and not previously realised in dogs; and obesity was found to further increase BOAS risk. This study provides evidence that breeding for brachycephaly leads to an increased risk of BOAS in dogs, with risk increasing as the morphology becomes more exaggerated. As such, dog breeders and buyers should be aware of this risk when selecting dogs, and breeding organisations should actively discourage exaggeration of this high-risk conformation in breed standards and the show ring

Cycle lengths in sparse graphs

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that \Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdos and Gyarfas that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average degree of graph of order n with no cycle of length in the sequence, where \log^* n is the number of times the binary logarithm must be applied to n to get a number which is at mos