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The inapproximability for the (0,1)-additive number

Ahadi, A ; Sharif University of Technology | 2016

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  1. Type of Document: Article
  2. Publisher: Discrete Mathematics and Theoretical Computer Science , 2016
  3. Abstract:
  4. An additive labeling of a graph G is a function H: V(G)→ N, such that for every two adjacent vertices v and u of G, Σw∼v l(w) = Σw∼vl(w) (x ∼ y means that x is joined to y). The additive number of G, denoted by η(G), is the minimum number k such that G has a additive labeling l: V(G)→ Nk. The additive choosability of a graph G, denoted by ηl(G), is the smallest number k such that G has an additive labeling for any assignment of lists of size k to the vertices of G, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph G, η(G) = ηe(G). We give a negative answer to this conjecture and we show that for every k there is a graph G such that ηe(G) - η(G) ≥ k. A (0, 1)-additive labeling of a graph G is a function H: V(G)→ {0,1}, such that for every two adjacent vertices v and u of G, Σw∼vl(w) = Σw∼v l(w). A graph may lack any (0,1)-additive labeling. We show that it is NP-complete to decide whether a (0,1)-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph G with some (0,1)-additive labelings, the (0,1)-additive number of G is defined as σ1 (G) = minl∈Γ Σv∈ν l(v) where Γ is the set of (0,1)-additive labelings of G. We prove that given a planar graph that admits a (0, 1)-additive labeling, for all ϵ > 0, approximating the (0, 1)-additive number within n1-ϵ is NP-hard
  5. Keywords:
  6. 1)-Additive labeling ; Additive labeling ; Additive number ; Combinatorial mathematics ; Computational complexity ; Computer science ; 1)-Additive number ; Adjacent vertices ; Choosability ; Inapproximability ; Lucky number ; Perfect graph ; Planar graph ; Triangle-free graphs ; Graph theory
  7. Source: Discrete Mathematics and Theoretical Computer Science ; Volume 17, Issue 3 , 2016 , Pages 217-226 ; 14627264 (ISSN)
  8. URL: https://arxiv.org/abs/1306.0182