A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K 1,4 with all vertices having degree at most 11.
A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on n vertices has at most n − 2 crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product… (More)
The effects of humor on increasing discomfort thresholds were tested with Transcutaneous End Nerve Stimulation (TENS). Undergraduate students (n = 31) with high or low scores on Martin and Lecourt's Situational Humor Questionnaire were randomly assigned to a humor or nonhumor condition. Discomfort thresholds for TENS were assessed before and during… (More)
with the main center of interest being near Mirabel, Quebec. The AIRS II project operational objectives are to: a) develop techniques/systems to remotely detect, diagnose and forecast hazardous winter conditions at airports, b) improve weather forecasts of aircraft icing conditions, c) better characterize the aircraft-icing environment and d) improve our… (More)
A graph is called 1-planar if there exists a drawing in the plane so that each edge contains at most one crossing. We study maximal 1-planar graphs from the point of view of properties of their diagrams, local structure and hamiltonicity.