The girth of cubic graphs

• Main Author: Hoare, Miles Jonathan
• Published: Royal Holloway, University of London 1982
• Subjects: 511; Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.704504

We start with an account of the known bounds for n(3,g), the number of vertices in the smallest trivalent graph of girth g, for g ≤ 12, including the construction of the smallest known trivalent graph of girth 9. This particular graph has 58 vertices - the 32 known trivalent graphs with 60 vertices are also catalogued and in some cases constructed. We prove the existence of vertex transitive trivalent graphs of arbitrarily high girth using Cayley graphs. The same result is proved for symmetric (that is vertex transitive and edge transitive) graphs, and a family of 2-arctransitive graphs for which the girth is unbounded is exhibited. The excess of trivalent graphs of girth g is shown to be unbounded as a function of g. A lower bound for the number of vertices in the smallest trivalent Cayley graph of girth g is then found for all g ≤ 9, and in each case it is shown that this bound is attained. We also establish an upper bound for the girth of Cayley graphs of subgroups of Aff (p^f) the group of linear transformations of the form x → ax + b where a,b are members of the field with p^f elements and a is non-zero. This family contains the smallest known trivalent graphs of girth 13 and 14, which are exhibited. Lastly a family of 4-arctransitive graphs for which the girth may be unbounded is constructed using "sextets". There is a graph in this family corresponding to each odd prime, and the family splits into several subfamilies depending on the congruency class of this prime modulo 16. The graphs corresponding to the primes congruent to 3,5,11,13 modulo 16 are actually 5-arctransitive. The girth of many of these graphs has been computed and graphs with girths up to and including 32 have been found.