Computational determination of (3,11)(3,11) and (4,7)(4,7) cages

A (k,g)(k,g)-graph is a k-regular graph of girth g  , and a (k,g)(k,g)-cage   is a (k,g)(k,g)-graph of minimum order. We show that a (3,11)(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)(4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders of (3,13)(3,13)-cages and (3,14)(3,14)-cages to 202 and 260, respectively. The methods used were a combination of heuristic hill-climbing and an innovative backtrack search.