Abstract
The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as L2,1 labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment.The Distance Edge Labeling problem asks whether the edges of a given graph can be labeled such that the labels of adjacent edges differ by at least two and the labels of edges at distance two differ by at least one. Labels are chosen from the set {0,1,…,λ} for λ fixed.We present a full classification of its computational complexity—a dichotomy between the polynomial-time solvable cases and the remaining cases which are NP-complete. We characterize graphs with λ≤4 which leads to a polynomial-time algorithm recognizing the class and we show NP-completeness for λ≥5 by several reductions from Monotone Not All Equal 3-SAT.Moreover, there is an absolute constant c>0 such that there is no 2cn-time algorithm deciding the Distance Edge Labeling problem unless the exponential time hypothesis fails.
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