On regular graphs of girth six arising from projective planes

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q >= k. Examining the construction method, we prove that starting from PG(2. q), q = p(h), p prime, there are no other constructions using this idea resulting in a (q + 1 - t)-regular graph of girth six than the known ones, if t is not too large (t <= p and roughly t < q(1/6)/8). Both algebraic and combinatorial tools are used. (c) 2012 Elsevier Ltd. All rights reserved.