Abstract

We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g)- cages, the minimum order 3-regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)-cages, and we were able to confirm that (3; 11)-cages have order 112 for the first time ever. The lower bound for a (3; 13)-cage is improved from 196 to 202 using the same approach. Also, we determined that a (3; 14)-cage has order at least 258. 1 Cages In this paper, we consider finite undirected graphs. Any undefined notation follows Bondy and Murty [7]. The girth of a graph is the size of a smallest cycle. A (r; g)- cage is an r-regular graph of minimum order with girth g. It is known that (r; g)-cages always exist [11]. Some nice pictures of small cages are given in [9, pp. 54-58]. The classification of the cages has attracted much interest amongst the graph theory community, and many of these have special nam...

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