Girth (tack) slightly too small, a girth extender may be used. A girth extender attaches to the billets of the saddle and lengthens them, so that a shorter girth may be used. Most jumping saddles have three billets. This not only allows the rider a spare should one break, but can also provide an adjustment option. For horses on which the saddle sits nicely, neither slipping forward or back, the first and third billets should be used. On horses where the saddle slips back, the first and second billets should be used. The second and third should never be used together, as
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