## Fast Backtracking Principles Applied to Find New Cages (1998)

Venue: | Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Mckay98fastbacktracking,

author = {Brendan Mckay and Wendy Myrvold and Jacqueline Nadon},

title = {Fast Backtracking Principles Applied to Find New Cages},

booktitle = {Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA},

year = {1998},

pages = {188--191}

}

### OpenURL

### 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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Citation Context ...02 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... |

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Citation Context ...od compromise is to perform strong re2 Girth LB Extra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph =-=[19]-=- 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 [1] [17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of ... |

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Citation Context ...graph theory community, and many of these have special names. Our interest in cages arose from the problem of determining the structure of the most reliable networks under the all-terminal model (see =-=[10]-=- for a survey of network reliability, and [6] for an introduction to the synthesis question). We suspect that large girth is a critical factor in maximizing the reliability when edges are very reliabl... |

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Citation Context ...e numbered sequentially starting with x and terminating at a precomputed vertex number end iso[x]. For example, if the desired girth is seven, the values of end iso are as pictured in Figure 2. Nauty =-=[14]-=-, a general program for graph isomorphism, was used to aid in the detection and elimination near the root of the search tree of branches which would not result in new cages. The ideas we applied are m... |

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Citation Context ...ation 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 names. O... |

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Citation Context ...graphs have an even number of vertices due to the elementary observation that a graph must have an even number of vertices of odd degree. 1. What we know. An obvious lower bound (given for example in =-=[4]-=-) on the order of a (3; g)-cage is (g) = ae 2 (g+2)=2 \Gamma 2; if g is even; and 3 \Theta 2 (g\Gamma1)=2 \Gamma 2; if g is odd. This is because a breadth first search tree truncated to height b(g \Ga... |

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Citation Context ... be extremely successful in reducing the size of the search tree but can greatly increase the work done for each node. A good compromise is to perform strong re2 Girth LB Extra Found Opt. Cages 3 4 0 =-=[18]-=- K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 [1] ... |

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Citation Context ...ical applications) and give evidence to support this for the 3-regular case [15, 16]. We concentrate on the 3-regular cages; a complete history of the results for this and other cases can be found in =-=[20]-=-. We summarize the status of the problem for girths three through twelve in Figure 1. The lower bound (LB) on the order is more fully explained in Section 3. The order of the cage equals the lower bou... |

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Citation Context ... in a straightforward manner. 7. Work distribution. For girth 9, the program was fast enough to be run on a single computer. For larger girth, subproblems were automatically distributed using autoson =-=[13]-=-. Hence we were able to obtain several years of computer time in just a couple of weeks. 4 Conclusions and Future Research Computer validation of mathematical results is still a relatively new phenome... |

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Citation Context ... special names. Our interest in cages arose from the problem of determining the structure of the most reliable networks under the all-terminal model (see [10] for a survey of network reliability, and =-=[6]-=- for an introduction to the synthesis question). We suspect that large girth is a critical factor in maximizing the reliability when edges are very reliable (the situation most often occurring in prac... |

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Citation Context ... K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 [1] =-=[17]-=- 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only simple fast tests at other ... |

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Citation Context ...K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 [1] [17] 3 [21] 11 94 18 =-=[2]-=- NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only simple fast tests at other places. 3. If there ... |

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Citation Context ...d (Found) but only later was it confirmed that they had minimum order (Opt.). The number of cages up to isomorphism is indicated in the last column (Cages). For girth 13, Brinkmann, McKay, and Saager =-=[8]-=- Australian National University. y University of Victoria. Supported by NSERC. z Sandwell Construction indicated a lower bound of 196 vertices. We used our program to verify that at least 202 vertices... |

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Citation Context ...[18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 =-=[1]-=- [17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only simple fast tests at o... |

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Citation Context ...tra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 =-=[22, 5]-=- [8] 18 [8] 10 62 8 [1] [17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only... |

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Citation Context ...is a critical factor in maximizing the reliability when edges are very reliable (the situation most often occurring in practical applications) and give evidence to support this for the 3-regular case =-=[15, 16]-=-. We concentrate on the 3-regular cages; a complete history of the results for this and other cases can be found in [20]. We summarize the status of the problem for girths three through twelve in Figu... |

3 |
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Citation Context ...is a critical factor in maximizing the reliability when edges are very reliable (the situation most often occurring in practical applications) and give evidence to support this for the 3-regular case =-=[15, 16]-=-. We concentrate on the 3-regular cages; a complete history of the results for this and other cases can be found in [20]. We summarize the status of the problem for girths three through twelve in Figu... |

2 |
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Citation Context ...r each node. A good compromise is to perform strong re2 Girth LB Extra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 =-=[12]-=- McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 [22, 5] [8] 18 [8] 10 62 8 [1] [17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests ... |

1 |
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Citation Context ...tra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte-Coexter graph [18] 9 46 12 =-=[22, 5]-=- [8] 18 [8] 10 62 8 [1] [17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3-regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only... |