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On the uniqueness of d-vertex magic constant

100%

Arumugam S., Kamatchi N., Vijayakumar G. R.

Discussiones Mathematicae Graph Theory

2014

tom 34

nr 2

279-286

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.

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Fractional global domination in graphs

100%

Arumugam S., Karuppasamy K., Hamid I.

Discussiones Mathematicae Graph Theory

2010

tom 30

nr 1

33-44

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g(N[v]) = ∑_{u ∈ N[v]}g(u) ≥ 1$ and $g(\overline{N(v)}) = ∑_{u ∉ N(v)}g(u) ≥ 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{fg}(G)$ is defined as follows: $γ_{fg}(G)$ = min{|g|:g is an MGDF of G } where $|g| = ∑_{v ∈ V} g(v)$. In this paper we initiate a study of this parameter.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Arumugam S.

1 Hamid I.

1 Kamatchi N.

1 Karuppasamy K.

1 Vijayakumar G. R.

1 2014

1 2010

Wyniki wyszukiwania

On the uniqueness of d-vertex magic constant

Fractional global domination in graphs