64 research outputs found
On Distance Magic Harary Graphs
This paper establishes two techniques to construct larger distance magic and
(a, d)-distance antimagic graphs using Harary graphs and provides a solution to
the existence of distance magicness of legicographic product and direct product
of G with C4, for every non-regular distance magic graph G with maximum degree
|V(G)|-1.Comment: 12 pages, 1 figur
Regular graphs of odd degree are antimagic
An antimagic labeling of a graph with edges is a bijection from
to such that for all vertices and , the sum of
labels on edges incident to differs from that for edges incident to .
Hartsfield and Ringel conjectured that every connected graph other than the
single edge has an antimagic labeling. We prove this conjecture for
regular graphs of odd degree.Comment: 5 page
Proof of a local antimagic conjecture
An antimagic labelling of a graph is a bijection
such that the sums
distinguish all vertices. A well-known conjecture of Hartsfield and Ringel
(1994) is that every connected graph other than admits an antimagic
labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \&
Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \&
Lyngsie (2017)) independently introduced the weaker notion of a local antimagic
labelling, where only adjacent vertices must be distinguished. Both sets of
authors conjectured that any connected graph other than admits a local
antimagic labelling. We prove this latter conjecture using the probabilistic
method. Thus the parameter of local antimagic chromatic number, introduced by
Arumugam et al., is well-defined for every connected graph other than .Comment: Final version for publication in DMTCS. Changes from previous version
are formatting to journal style and correction of two minor typographical
error
Distance Magic Graphs - a Survey
Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V &rarr; {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>&Sigma; f(u) = k </i> for all <i>v &epsilon; V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-2
Approximate results for rainbow labelings
Article de recercaPreprin
A SURVEY OF DISTANCE MAGIC GRAPHS
In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x)
where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph.
In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs.
In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants.
In Chapter 3, we study graph labelings that retain the same labels as distance magic labelings, but alter the definition in some other way. These labelings include balanced distance magic labelings, closed distance magic labelings, D-distance magic labelings, and distance antimagic labelings.
In Chapter 4, we examine results on neighborhood magic labelings, group distance magic labelings, and group distance antimagic labelings. These graph labelings change the label set, but are otherwise similar to distance magic graphs.
In Chapter 5, we examine some applications of distance magic and distance antimagic labeling to the fair scheduling of tournaments.
In Chapter 6, we conclude with some open problems
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