On ( a , d )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs
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ON ( a , d ) -DISTANCE ANTI-MAGIC AND 1-VERTEX BIMAGIC VERTEX LABELINGS OF CERTAIN TYPES OF GRAPHS
M. F. Semeniuta
UDC 519.17
Abstract. The results for the corona Pn o P1 are generalized, which make it possible to state that Pn o P1 is not an ( a, d ) -distance antimagic graph for arbitrary values of a and d . A condition for the existence of an ( a, d ) -distance antimagic labeling of a hypercube Q n is obtained. Functional dependencies are found that generate this type of labeling for Q n . It is proved by the method of mathematical induction that Q n is a (2 n + n – 1, n - 2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G È G is established. Keywords: distance magic labeling, ( a, d ) -distance antimagic labeling, 1-vertex bimagic vertex labeling, n -dimensional cube, corona. INTRODUCTION In recent years, graph labelings are increasingly becoming the subject of intensive investigations [1]. A stimulus to the development of this direction and accumulation of theoretical results is the presence of practical problems in different fields of activity [2–7]. A labeling of a graph G = (V , E ) is understood to be a mapping that associates elements of the graph with numbers belonging to a given set according to a definite rule. A vertex, edge, or total labeling is considered depending on the domain of its definition. The way of computing the weight of a vertex or an edge depends on the type of labeling. If all weights are the same, then the magic type of labeling is obtained, and if all weights are different, then the antimagic type is obtained. If the weights of vertices form an arithmetic progression whose first term is a and difference is d , then we have a vertex ( a, d ) -distance antimagic labeling. It was first proposed by S. Arumugam and N. Kamatchi in 2012 [8]. This article continues the investigations described in [7–11]. In [8], the first general results are presented and open problems are formulated whose partial solution is presented in [7–12]. The description of characteristics of ( a, d )-distance antimagic regular graphs, chains, and constructions of graphs obtained with the help of one-place and two-place operations over given graphs started in [7, 8]. S. Arumugam and N. Kamatchi constructed an ( n + 2, 1)-distance antimagic labeling of the Cartesian product of a cycle C n and the graph K 2 [8], and D. Froncek proved that the disjunctive union of copies of the Cartesian product of two complete graphs and their complement is an ( a, 2) -distance antimagic graph and an ( a, 1)-distance antimagic graph, respectively [7]. D. Froncek also showed that disjunctive copies of the hypercube Q 3 form an ( a, 1) -distance antimagic graph. In [10, 11], the conditions for the existence of an ( a, d ) -distance antimagic labeling of circulant graphs and friendship graphs are found and theorems that extend families of not ( a, d ) -distance
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