Fast backtracking principles applied to find new cages
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
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A (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.