Locating-dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locating-dominating codes in paths P"n. They conjectured that if r>=2 is a fixed integer, then the smallest cardinality of an r-locating-dominating code in P"n, denoted by M"r^L^D(P"n), satisfies M"r^L^D(P"n)[email protected]?(n+1)/[email protected]? for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r>=3 we have M"r^L^D(P"n)[email protected]?(n+1)/[email protected]? for all n>=n"r when n"r is large enough. In addition, we solve a conjecture on location-domination with segments of even length in the infinite path.