We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph , written , is the minimum number of cops needed to ensure the robber's capture. We present bounds on the localization number of incidence graphs of balanced incomplete block designs. Exact values of the localization number are given for the incidence graphs of projective and affine planes. Bounds are given for Steiner systems and for transversal designs.

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We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is n + a if and only if the maximum size of a partial transversal is either n − 2a or n − 2a + 1. (2) A minimal cover in a Latin square of order n has size at most μ n = 3(n + 1/2 − √ n + 1/4). (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to μ n . (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to μ n . (5) If 1 k n/2 and n 5 then there is a Latin square of order n with a maximal partial transversal of size n − k. (6) For any ε > 0, asymptotically almost all Latin squares have no maximal partial transversal of size less than n − n 2/3+ε .

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