Partial linear spaces and identifying codes

Let (P ,L, I) be a partial linear space and X ⊆ P ∪ L. Let us denote by (X)I = ⋃ x∈X {y : yIx} and by [X] = (X)I ∪ 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 ⊆ P ∪ L with |X| ≤ 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) + k points and 2(k − 1) + k lines whose incidence graphs do not admit a (1,≤ k)-identifying code. Finally, we show that the smallest (k; 6)-graphs known up to now for k − 1 not a prime power admit a (1,≤ k)-identifying code. [email protected] [email protected] [email protected] [email protected]