On critical subgraphs of colour-critical graphs

AbstractSome results on the distribution of critical subgraphs in colour-critical graphs are obtained. Characterizations of k-critical graphs in terms of their (k − 1)-critical subgraphs are given. The special case k = 4 is considered, and it is proved that if a 4-critical graph I′ has a vortex x of large valency (compared to the number of vertices of I′ not adjacent to x), then I′ contains vertices of valency 3. Finally, a list of all 4-critical graphs with ≤ 9 vertices is exhibited

On a special case of Hadwiger's conjecture

Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case

Socioeconomic status and risk of incident venous thromboembolism

Background: Although venous thromboembolism (VTE) is a leading cause of morbidity and mortality, and socioeconomic status (SES) affects human health and health behavior, few studies have examined the association between SES and VTE. Objectives: We aimed to investigate the association between SES, assessed individually and in a composite score by levels of education, income, and employment status, and incident VTE. Methods: We used Danish national registries to identify 51 350 persons aged 25– 65 years with incident VTE during 1995–2016. For each case, we used incidence density sampling to select five age-, sex-, and index-year-matched controls from the general Danish population (n = 256 750). SES indicators, including education, income, and employment status, were assessed 1 and 5 years before the VTE. We used conditional logistic regression to compute odds ratios (ORs) with 95% confidence intervals (CIs) for VTE according to individual SES indicators and a composite SES score in analyses adjusted for age, sex, and comorbidities. Results: Compared with low levels, high educational level (OR 0.74; 95% CI 0.71– 0.77), high income (OR 0.70; 95% CI 0.68–0.72), and high employment status (OR 0.66; 95% CI 0.64–0.68) were associated with decreased risk of VTE, even after adjusting for comorbidities. A composite SES score was superior to the individual indicators in assessing VTE risk (OR for high vs. low score: 0.61; 95% CI 0.59–0.63). In sensitivity analysis with SES indicators measured 5 years before the VTE, the risk estimates remained essentially the same. Conclusion: High levels of both individual SES indicators and a composite SES score were associated with decreased VTE risk