Cycle-magic graphs

AbstractA simple graph G=(V,E) admits a cycle-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given cycle C. Then the graph G is C-magic if there exists a total labelling f:V∪E→{1,2,…,|V|+|E|} such that, for every subgraph H′=(V′,E′) of G isomorphic to C, ∑v∈V′f(v)+∑e∈E′f(e) is constant. When f(V)={1,…,|V|}, then G is said to be C-supermagic.We study the cyclic-magic and cyclic-supermagic behavior of several classes of connected graphs. We give several families of Cr-magic graphs for each r⩾3. The results rely on a technique of partitioning sets of integers with special properties

Magic coverings and the Kronecker product

In this paper we study a relationship existing among (super) magic coverings and the well known Kronecker product of matrices. We also introduce the concept of Zn–property for digraphs in order to study this relation mentioned before. The results obtained in this paper can also be applied to construct S–magic partitions.Preprin

Vertex Magic Group Edge Labelings

A project submitted to the faculty of the graduate school of the University of Minnesota in partial fulfillment of the requirements for the degree of Master of Science. May 2017. Major: Mathematics and Statistics. Advisor: Dalibor Froncek. 1 computer file (PDF); vi, 46 pages, appendix A, Ill. (some col.)A vertex-magic group edge labeling of a graph G(V;E) with |E| = n is an injection from E to an abelian group ᴦ of order n such that the sum of labels of all incident edges of every vertex x ϵ V is equal to the same element µ ϵ ᴦ. We completely characterize all Cartesian products Cn□Cm that admit a vertex-magic group edge labeling by Z2nm, as well as provide labelings by a few other finite abelian groups

H-E-Super Magic Decomposition of Graphs

An H-magic labeling in an H-decomposable graph G is a bijection f:V(G) U E(G) --> {1,2, … ,p+q} such that for every copy H in the decomposition, $\sum\limits_{v\in V(H)} f(v)+\sum\limits_{e\in E(H)} f(e)$ is constant. The function f is said to be H-E-super magic if f(E(G)) = {1,2, … ,q}. In this paper, we study some basic properties of m-factor-E-super magic labelingand we provide a necessary and sufficient condition for an even regular graph to be 2-factor-E-super magic decomposable. For this purpose, we use Petersen\u27s theorem and magic squares

Supermagic graphs with many different degrees

Let G = (V, E) be a graph with n vertices and e edges. A supermagic labeling of G is a bijection f from the set of edges E to a set of consecutive integers {a, a + 1,..., a + e - 1} such that for every vertex v is an element of V the sum of labels of all adjacent edges equals the same constant k. This k is called a magic constant of f, and G is a supermagic graph. The existence of supermagic labeling for certain classes of graphs has been the scope of many papers. For a comprehensive overview see Gallian's Dynamic survey of graph labeling in the Electronic Journal of Combinatorics. So far, regular or almost regular graphs have been studied. This is natural, since the same magic constant has to be achieved both at vertices of high degree as well as at vertices of low degree, while the labels are distinct consecutive integers.Web of Science4141050104

Methods for constructing magic graph labelings

V této diplomové práci se zaměřujeme na distančně magické ohodnocení, handicapové ohodnocení, k-handicapové ohodnocení, distančně antimagické ohodnocení a supermagické ohodnocení grafu. Je zde zpracován přehled známých výsledků a nutné podmínky existence pro každé ze zmíněných ohodnocení. Na základě známých výsledků bylo cílem navrhnout analogické postupy i pro další typy ohodnocení. Navíc jsme se pokusili postupy aplikovat. V práci zkoušíme kombinovat grafy s různými ohodnoceními tak, aby měl výsledný graf jedno z daných ohodnocení. Hledáme podobné a odlišné vlastnosti ohodnocení, které mohou být faktorem existence ohodnocení výsledného grafu. Pro supermagické ohodnocení jsme uvedli tzv. spektrum magické konstanty, kde pro jeden pravidelný graf můžeme zvětšením všech labelů grafu o stejnou hodnotu získat nekonečně mnoho nových supermagických ohodnocení. U k-handicapového ohodnocení jsme ukázali tvrzení, díky kterého můžeme pomocí k-dimenzionální magické krychle odvodit (-k)-handicapové ohodnocení grafu. Dalším výsledkem dosaženým v této práci je konstrukce supermagického ohodnocení pro graf, který byl vytvořen sjednocením dvou supermagických grafů, pomocí posunutí původních ohodnocení sjednocovaných grafů. Na konci práce je uvedeno shrnutí, kde zkoumáme všechny použité konstrukce a pro jaký druh ohodnocení a typ grafu lze tato konstrukce použít.In this master thesis we focus on distance magic labeling, handicap labeling, k-handicap labeling, distance antimagic labeling and supermagic labeling. The known results and necessary conditions for existence of these labelings are compiled. Based on the known results, the goal was to come up with similar approaches for other labelings as well. We try to combine graphs with different labelings so that the resulting graph has one of the given labelings. We are looking for similar and different properties of the labelings, that could be a reason for existence of the labelings of the resulting graph. For supermagic labeling, we introduced the so-called spectrum of the magic constant, where we can increase all labels by the same value to get an infinite number of new supermagic labelings for one regular graph. We also came up with a proposition that allows us to use a k-dimensional magic cube as a (-k)-handicap labeling of a graph. Another outcome of this thesis is the construction of supermagic labeling of a graph, that was created by union of two supermagic graphs, by shifting the original labeling of the unified graphs. A summary is given at the end of the thesis, where we examine all the constructions used and for what kind of labeling and type of graphs these constructions can be used.470 - Katedra aplikované matematikyvýborn