Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

Denning in brown bears

Hibernation represents an adaptation for coping with unfavorable environmental conditions. For brown bears Ursus arctos, hibernation is a critical period as pronounced temporal reductions in several physiological functions occur. Here, we review the three main aspects of brown bear denning: (1) den chronology, (2) den characteristics, and (3) hibernation physiology in order to identify (a) proximate and ultimate factors of hibernation as well as (b) research gaps and conservation priorities. Den chronology, which varies by sex and reproductive status, depends on environmental factors, such as snow, temperature, food availability, and den altitude. Significant variation in hibernation across latitudes occurs for both den entry and exit. The choice of a den and its surroundings may affect individual fitness, for example, loss of offspring and excessive energy consumption. Den selection is the result of broad‐ and fine‐scale habitat selection, mainly linked to den insulation, remoteness, and availability of food in the surroundings of the den location. Hibernation is a metabolic challenge for the brown bears, in which a series of physiological adaptations in tissues and organs enable survival under nutritional deprivation, maintain high levels of lipids, preserve muscle, and bone and prevent cardiovascular pathologies such as atherosclerosis. It is important to understand: (a) proximate and ultimate factors in denning behavior and the difference between actual drivers of hibernation (i.e., factors to which bears directly respond) and their correlates; (b) how changes in climatic factors might affect the ability of bears to face global climate change and the human‐mediated changes in food availability; (c) hyperphagia (period in which brown bears accumulate fat reserves), predenning and denning periods, including for those populations in which bears do not hibernate every year; and (d) how to approach the study of bear denning merging insights from different perspectives, that is, physiology, ecology, and behavior.During this research, EGB was financially supported by a FPU grant (FPU15‐03429) of the Spanish Ministry of Science, Innovation and Universities. V.P. was financially supported by: (1) the Excellence Project CGL2017‐82782‐P financed by the Spanish Ministry of Science, Innovation and Universities, the Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER, EU); and (2) a GRUPIN research grant from the Regional Government of Asturias (Ref.: IDI/2018/000151).Peer reviewe