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We study three invariants of graphs measuring how far a graph is from having a proper 3-edge-coloring. We show that they have the same value on certain classes of graphs, in particular on the class of cubic graphs.
Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locating–dominating codes in paths Pn. They conjectured that if r≥2 is a fixed integer, then the smallest cardinality of an r-locating–dominating code in Pn, denoted by MrLD(Pn), satisfies MrLD(Pn)=⌈(n+1)/3⌉ for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that...
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