The girth of cubic graphs

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) thegroup of linear transformations of the form x -> ax + b where a,b are members of the field with p elements and a is non-zero. This family contains thesmallest 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 3,5,11,13modulo 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.<p

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update