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Magic and supermagic dense bipartite graphs

Jaroslav Ivanco — 2007

Discussiones Mathematicae Graph Theory

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we prove that any balanced bipartite graph with minimum degree greater than |V(G)|/4 ≥ 2 is magic. A similar result is presented for supermagic regular bipartite graphs.

Decompositions of multigraphs into parts with the same size

Jaroslav Ivanco — 2010

Discussiones Mathematicae Graph Theory

Given a family ℱ of multigraphs without isolated vertices, a multigraph M is called ℱ-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of ℱ. We present necessary and sufficient conditions for existence of such decompositions if ℱ consists of all multigraphs of size q except for one. Namely, for a multigraph H of size q we find each multigraph M of size kq, such that every partition of the edge set of M into parts of cardinality q contains a part...

Supermagic Generalized Double Graphs 1

Jaroslav Ivančo — 2016

Discussiones Mathematicae Graph Theory

A graph G is called supermagic if it admits a labelling of the edges by pairwise di erent consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we will introduce some constructions of supermagic labellings of some graphs generalizing double graphs. Inter alia we show that the double graphs of regular Hamiltonian graphs and some circulant graphs are supermagic.

Note on independent sets of a graph

Jaroslav Ivančo — 1994

Mathematica Bohemica

Let the number of k -element sets of independent vertices and edges of a graph G be denoted by n ( G , k ) and m ( G , k ) , respectively. It is shown that the graphs whose every component is a circuit are the only graphs for which the equality n ( G , k ) = m ( G , k ) is satisfied for all values of k .

On supermagic regular graphs

Jaroslav Ivančo — 2000

Mathematica Bohemica

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.

Supermagic Graphs Having a Saturated Vertex

Jaroslav IvančoTatiana Polláková — 2014

Discussiones Mathematicae Graph Theory

A graph is called supermagic if it admits a labeling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we establish some conditions for graphs with a saturated vertex to be supermagic. Inter alia we show that complete multipartite graphs K1,n,n and K1,2,...,2 are supermagic.

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef BuckoJaroslav Ivančo — 1999

Discussiones Mathematicae Graph Theory

Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph G [ V i ] has the property i . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree....

Total edge irregularity strength of trees

Jaroslav IvančoStanislav Jendrol' — 2006

Discussiones Mathematicae Graph Theory

A total edge-irregular k-labelling ξ:V(G)∪ E(G) → {1,2,...,k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every...

On magic joins of graphs

Jaroslav IvančoTatiana Polláková — 2012

Mathematica Bohemica

A graph is called magic (supermagic) if it admits a labeling of the edges by pairwise different (and consecutive) integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we characterize magic joins of graphs and we establish some conditions for magic joins of graphs to be supermagic.

Decompositions of multigraphs into parts with two edges

Jaroslav IvančoMariusz MeszkaZdzisław Skupień — 2002

Discussiones Mathematicae Graph Theory

Given a family 𝓕 of multigraphs without isolated vertices, a multigraph M is called 𝓕-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of 𝓕. We present necessary and sufficient conditions for the existence of such decompositions if 𝓕 comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.

M 2 -Edge Colorings Of Cacti And Graph Joins

Július CzapPeter ŠugerekJaroslav Ivančo — 2016

Discussiones Mathematicae Graph Theory

An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.

Note on Petrie and Hamiltonian cycles in cubic polyhedral graphs

Jaroslav IvančoStanislav JendroľMichal Tkáč — 1994

Commentationes Mathematicae Universitatis Carolinae

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.ei̇t is also a Petrie cycle. The Petrie Hamiltonian cycle in an n -vertex plane cubic graph can be recognized by an O ( n ) -algorithm.

On magic and supermagic line graphs

Jaroslav IvančoZ. LastivkováA. Semaničová — 2004

Mathematica Bohemica

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We characterize magic line graphs of general graphs and describe some class of supermagic line graphs of bipartite graphs.

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