Surcingle longe lines, requires the use of a surcingle to thread the longe lines through the rings. Compared to a saddle, a surcingle allows more precise adjustment of side reins due to the placement of additional rings. While a saddle only provides one height to attach the rings (the girth buckles), and can be uneven or at the wrong position, a training surcingle places rings at more appropriate locations for ground work. Many surcingle designs allow the side reins to be attached at several different heights along the sides of the horse. When used without a saddle, a surcingle sits just
Girth (graph theory) repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number. Paul Erdős was the first to prove the general result, using the probabilistic method. More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a