Hooley-ann, hoolian and houlihan must be the most discussed words in mudcat.
Originally the rope-throw was hooley-ann or houlian. The rodeo term (and party term) was houlihan.
Alice shows how terms change as original meanings or applications are changed, at least in her region (Montana), and perhaps houlihan is becoming the word applied for the rope throw generally.

Here are definitions with source given.
hoolihan the act of leaping forward and alighting on the horns of a steer in order to knock the steer down. Jules Verne Allen, 1933, Cowboy Lore; Ramon F. Adams, 1944, Western Words.
Adams adds: "Also to throw a big time in town- to paint the town red."

hooley-ann (hoolian) a roping term, the throw used to to rope horses (and also to catch calves out of a bunch). [The loop is first thrown backward, it is a backhand loop but the rider rolls his wrist and the loop rolls over and flattens out]. Adams says the loop could be thrown thirty feet; it "is a fast loop and is strictly a head catch, being especially used to catch horses in a corral." Ramon F. Adams, 1936, Cowboy Lingo and Western Words, 1944.
It is thrown "with a rather small loop and has the additional virtue of landing with the honda sliding down the rope, taking up the slack as it goes (W. M. French, 1940, Ropes and Roping, Cattleman, XXVI, No. 12, 17-30; quoted in Adams).   

Dogie. From dogal, the vaquero term for a motherless calf. Donald Chavez y Gilbert, Cowboys - Vaqueros, Origins of the first American Cowboys. Chapter 9, Vaquero/Cowboy Lingo.
Cowboys vaqueros
An invaluable source.