Regular parcial linear spaces admitting (1;≤ k)-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. That is, k-regular bipartite graphs of girth at least six admitting a set C of vertices such that the setsN [x]∩C are nonempty and pairwise distinct for all vertex x ∈ X where X is a subset of vertices of |X | ≤ k. Moreover, we present a family of k-regular partial linear spaces on 2(k−1)2+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.