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Zero-sum magic labelings and null sets of regular graphs

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. Abstract:
  3. For every h ∈ ℕ, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l: E(G) → ℤh{0} such that the induced vertex labeling s: V (G) → ℤh, defined by s(v) = Puv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h ∈ ℕ such that G admits a zero-sum h-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if G is an r-regular graph, then for even r (r > 2), N(G) = ℕ and for odd r (r ≠ 5), ℕ {2, 4} ⊆ N(G). Moreover, we prove that if r is odd and G is a 2-edge connected r-regular graph (r ≠ 5), then N(G) = ℕ {2}. Also, we show that if G is a 2-edge connected bipartite graph, then ℕ {2, 3, 4, 5} ⊆ N(G)
  4. Keywords:
  5. Bipartite graph ; Magic labeling ; Null set ; Regular graph ; Zero-sum flows
  6. Source: Electronic Journal of Combinatorics ; Vol. 21, issue. 2 , May , 2014 ; ISSN: 10778926
  7. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p17