Partial linear spaces and identifying codes

Let ( P , L , I ) be a partial linear space and X ⊆ P ∪ L . Let us denote ( X ) I = ⋃ x ∈ X { y : y I x } and [ 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 ) 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.