Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nu_t(G)$ and $\tau_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\tau_t(G)/\nu_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nu_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $\nu_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

Patient preference and acceptability of calcium plus vitamin D3 supplementation: a randomised, open, cross-over trial

Preference for a drug formulation is important in adherence to long-term medication for chronic illnesses such as osteoporosis. We investigated the preference for and acceptability of chewable tablet containing calcium and vitamin D (Calci Chew D3, Nycomed) compared to that of a sachet containing calcium and vitamin D3 (Cad, Will-Pharma). This open, randomised, cross-over trial was set up to compare the preference and acceptability of two calcium plus vitamin D3 formulations (both with 500 mg calcium and 400/440 IU vitamin D3), given twice a day in patients with osteoporosis. Preference and acceptability were assessed by means of questionnaires. Preference was determined by asking the question, which treatment the patient preferred, and acceptability was measured by scoring five variables, using rating scales. Of the 102 patients indicating a preference for a trial medication, 67% preferred the chewable tablet, 19% the sachet with calcium and vitamin D3, and 15% stated no preference. The significant preference for Calci Chew D3 (p < 0.0001) was associated with higher scores for all five acceptability variables. The two formulations were tolerated equally well. A significant greater number of patients considered the chewable tablet as preferable and acceptable to the sachet, containing calcium and vitamin D3. Trial registration: Current Controlled Trials ISRCTN18822358

Optimal Use of Vitamin D When Treating Osteoporosis

Inadequate serum 25-hydroxyvitamin D (25[OH]D) concentrations are associated with muscle weakness, decreased physical performance, and increased propensity in falls and fractures. This paper discusses several aspects with regard to vitamin D status and supplementation when treating patients with osteoporosis in relation to risks and prevention of falls and fractures. Based on evidence from literature, adequate supplementation with at least 700 IU of vitamin D, preferably cholecalciferol, is required for improving physical function and prevention of falls and fractures. Additional calcium supplementation may be considered when dietary calcium intake is below 700 mg/day. For optimal bone mineral density response in patients treated with antiresorptive or anabolic therapy, adequate vitamin D and calcium supplementation is also necessary. Monitoring of 25(OH)D levels during follow-up and adjustment of vitamin D supplementation should be considered to reach and maintain adequate serum 25(OH)D levels of at least 50 nmol/L, preferably greater than 75 nmol/L in all patients

Identification of Natural Bispecific Antibodies against Cyclic Citrullinated Peptide and Immunoglobulin G in Rheumatoid Arthritis

BACKGROUND: Previous studies indicate that natural bispecific antibodies can be readily produced in vivo when the body is simultaneously stimulated with 2 distinct antigens. Patients with rheumatoid arthritis (RA) usually exhibit persistent immune responses to various autoantigens, raising the possibility that natural bispecific antibodies against 2 distinct autoantigens might exist. METHODOLOGY/PRINCIPAL FINDINGS: We identified the presence of natural bispecific antibodies against cyclic citrullinated peptide (CCP) and immunoglobulin G (IgG) in RA patients' sera by means of a double-antigen sandwich enzyme-linked immunosorbent assay (ELISA). The spontaneous emergence of bispecific antibodies was confirmed by mixing different proportions of 1 anti-CCP-positive serum and 1 rheumatoid factor (RF)-positive serum in vitro. Among the tested samples, positive correlations were found between the presence of bispecific antibodies and both IgG4 anti-CCP antibodies and IgG4 RF (r = 0.507, p<0.001 and r = 0.249, p = 0.044, respectively), suggesting that the IgG4 subclass is associated with this phenomenon. Furthermore, bispecific antibodies were selectively generated when several anti-CCP- and RF-positive sera were mixed pairwise, indicating that factors other than the monospecific antibody titers may also contribute to the production of the natural bispecific antibodies. CONCLUSIONS/SIGNIFICANCE: We successfully identified the presence of natural bispecific antibodies. Our results suggest that these antibodies originate from anti-CCP and RF in the sera of RA patients. The natural occurrence of bispecific antibodies in human diseases may provide new insights for a better understanding of the diseases. Further investigations are needed to elucidate their precise generation mechanisms and explore their clinical significance in disease development and progression in a larger study population