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

Computational determination of (3,11) and (4,7) cages

A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a (k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by Balaban in 1973 is minimal and unique. We also show that the order of a (4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The methods used were a combination of heuristic hill-climbing and an innovative backtrack search

Hyperbolic generalized triangle groups, property (T) and finite simple quotients

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan\u27s property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first-known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, that is, groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201