Partial linear spaces and identifying codes

Let (P, L, I) be a partial linear space and X subset of P boolean OR L. Let us denote (X)(I) = U(x is an element of X) {y : y/x) and [X] = (X)(I) boolean OR X. With this terminology a partial linear space (P, L, I) is said to admit a (1, <= k)-identifying code if and only if the sets [X] are mutually different for all X subset of P boolean OR L with vertical bar X vertical bar <= k. In this paper we give a characterization of k-regular partial linear spaces admitting a (1, <= k)-identifying code. Equivalently, we give a characterization of k-regular bipartite graphs of girth at least six admitting a (1, <= k)-identifying code. Moreover, we present a family of k-regular partial linear spaces on 2(k - 1)(2)+k points and 2(k - 1)(2)+k lines whose incidence graphs do not admit a (1, <= k)-identifying code. Finally, we show that the smallest (k; 6)-graphs known up until now for k - 1 where k - 1 is not a prime power, admit a (1, <= k)-identifying code. (C) 2010 Elsevier Ltd. All rights reserved.