Exoo, GeoffreyMcKay, BrendanMyrvold, WendyNadon, Jacqueline2015-12-101570-8667http://hdl.handle.net/1885/57758A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a (k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by Balaban in 1973 is minimal and unique. We also show that the order of a (4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The methods used were a combination of heuristic hill-climbing and an innovative backtrack search.Keywords: Backtrack search; Cage; Girth; Hill climbing; Lower bounds; Minimum order; Regular graph; Regular graphs; Graph theory; Heuristic methods Cage; Girth; Regular graph201110.1016/j.jda.2010.11.0012016-02-24