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The Distance Magic Index of a Graph

100%
Godinho A., Singh T., Arumugam S.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
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nr 1
135-142
EN
Let G be a graph of order n and let S be a set of positive integers with |S| = n. Then G is said to be S-magic if there exists a bijection ϕ : V (G) → S satisfying ∑x∈N(u) ϕ(x) = k (a constant) for every u ∈ V (G). Let α(S) = max{s : s ∈ S}. Let i(G) = min α(S), where the minimum is taken over all sets S for which the graph G admits an S-magic labeling. Then i(G) − n is called the distance magic index of the graph G. In this paper we determine the distance magic index of trees and complete bipartite graphs.
2
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Constant Sum Partition of Sets of Integers and Distance Magic Graphs

100%
Cichacz S., Gőrlich A.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
97-106
EN
Let A = {1, 2, . . . , tm+tn}. We shall say that A has the (m, n, t)-balanced constant-sum-partition property ((m, n, t)-BCSP-property) if there exists a partition of A into 2t pairwise disjoint subsets A1, A2, . . . , At, B1, B2, . . . , Bt such that |Ai| = m and |Bi| = n, and ∑a∈Ai a = ∑b∈Bj b for 1 ≤ i ≤ t and 1 ≤ j ≤ t. In this paper we give sufficient and necessary conditions for a set A to have the (m, n, t)-BCSP-property in the case when m and n are both even. We use this result to show some families of distance magic graphs.
3
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Distance Magic Cartesian Products of Graphs

75%
Cichacz S., Froncek D., Krop E., Raridan C.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
299-308
EN
A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ : V → {1, . . . , n} such that the weight of every vertex v, computed as the sum of the labels on the vertices in the open neighborhood of v, is a constant. In this paper, we show that hypercubes with dimension divisible by four are not distance magic. We also provide some positive results by proving necessary and sufficient conditions for the Cartesian product of certain complete multipartite graphs and the cycle on four vertices to be distance magic.
4
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Union of Distance Magic Graphs

63%
Cichacz S., Nikodem M.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
nr 1
239-249
EN
A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ from V to the set {1, . . . , n} such that the weight w(x) = ∑y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.
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