Planar Hypohamiltonian Graphs on 40 Vertices

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles arereplaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices.The first two authors were supported by the Australian Research Council. The work of the third author was supported in part by the Academy of Finland under the Grant Nos. 132122 and 289002; the work of the fourth author was supported by Grant No. 132122, the GETA Graduate School, and the Nokia Foundation. The last author is a PhD fellow at Ghent University on the BOF (Special Research Fund) scholarship 01DI1015

Consumer rejection threshold, acceptability rates, physicochemical properties, and shelf‐life of strawberry‐flavored yogurts with reductions of sugar

BACKGROUND: There is an increasing demand for reduced-sugar products due to the worldwide prevalence of obesity, diabetes, and cardiovascular diseases. The aim of this study was to evaluate the effects of sugar (sucrose) reductions on the acceptability, preference, and quality of strawberry-flavored yogurts. A consumer rejection threshold test and an acceptability test (N = 53) were conducted using six yogurt samples with decreasing concentrations of sugar (12–5/100 g). Additional physicochemical tests (pH, °Brix, water-holding-capacity, viscosity, and color) were conducted to examine the quality and shelf-life of strawberry-flavored yogurts with reductions of sucrose during 28 days of storage at 4 °C. RESULTS: Reduction of sucrose affected the acceptability and physicochemical characteristics of yogurts. The consumer rejection threshold showed that sucrose in strawberry-flavored yogurts could be reduced to 5.25/100 g from an initial concentration of 12/100 g without affecting the preferences of consumers. The 71%-sucrose (8.50/100 g of yogurt) was perceived as the most liked (6.27 using a nine-point hedonic scale) and the most preferred (rank sum = 127.50) yogurt sample. For the physicochemical properties of yogurts, the viscosity (3263–5473 cP) decreased, and the color lightness (80.98–85.44) increased during 28 days of storage at 4 °C. CONCLUSION: Physicochemical properties and preferences were affected by the reduction of sugar. The consumer rejection threshold analysis showed that sucrose can be reduced to less than half of the initial concentration. These findings are useful to understand consumers' acceptability and shelf-life of yogurts with reduced-sugar formulations in the developing of new products