Abstract
The aim of this paper is to construct new small regular graphs with girth $7$ using integer programming techniques. Over the last two decades solvers for integer programs have become more and more powerful and have proven to be a useful aid for many hard combinatorial problems. Despite successes in many related fields, these optimisation tools have so far been absent in the quest for small regular graphs with a given girth. Here we illustrate the power of these solvers as an aid to construct small regular girth $7$ graphs from girth $8$ cages.
Highlights
IntroductionFor the cage problem one aims to construct regular graphs of degree k with minimum number of vertices required to have girth g
The girth of a graph is the length of its shortest cycle
Despite successes in many related fields, these optimisation tools have so far been absent in the quest for small regular graphs with a given girth
Summary
For the cage problem one aims to construct regular graphs of degree k with minimum number of vertices required to have girth g. In this paper we construct new record graphs with girth 7, improving the upper bounds for regular girth 7-graphs with degrees up to 14. This is achieved by a novel approach that uses the power of modern linear programming solvers and combinatoric properties of some known girth 8 graphs. For all other degrees and girths only record graphs, which give an upper bound on the order of the cages, are known.
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