13,906 research outputs found
Bounds on the size of super edge-magic graphs depending on the girth
Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic
graph then q 2p−3. Furthermore, if G is super edge-magic and q = 2p−3, then the girth
of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then
q 2p − 5. In this paper we show that there are infinitely many graphs which are super
edge-magic, have girth 5, and q = 2p−5. Therefore the maximum size for super edge-magic
graphs of girth 5 cannot be reduced with respect to the maximum size of super edge-magic
graphs of girth 4.Preprin
Recent studies on the super edge-magic deficiency of graphs
A graph is called edge-magic if there exists a bijective function
such that is a constant for each . Also,
is said to be super edge-magic if . Furthermore, the
super edge-magic deficiency of a graph is defined
to be either the smallest nonnegative integer with the property that is super edge-magic or if there exists no such integer
. In this paper, we introduce the parameter as the minimum
size of a graph of order for which all graphs of order and size at
least have , and provide
lower and upper bounds for . Imran, Baig, and
Fe\u{n}ov\u{c}\'{i}kov\'{a} established that for integers with , , where is the
cartesian product of the cycle of order and the complete graph
of order . We improve this bound by showing that when is even. Enomoto,
Llad\'{o}, Nakamigawa, and Ringel posed the conjecture that every nontrivial
tree is super edge-magic. We propose a new approach to attak this conjecture.
This approach may also help to resolve another labeling conjecture on trees by
Graham and Sloane
Perfect (super) Edge-Magic Crowns
A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|} such that the sum f(x)+f(xy)+f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G))={1,2,…,|V(G)|} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns Cm¿K¯¯¯¯¯n and we prove that the crowns are perfect (super) edge-magic when m=pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of Cm¿K¯¯¯¯¯n, in terms of the prime factors of m.Postprint (updated version
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
Super edge-magic deficiency of join-product graphs
A graph is called \textit{super edge-magic} if there exists a bijective
function from to such
that and is a
constant for every edge of . Furthermore, the \textit{super
edge-magic deficiency} of a graph is either the minimum nonnegative integer
such that is super edge-magic or if there exists no
such integer.
\emph{Join product} of two graphs is their graph union with additional edges
that connect all vertices of the first graph to each vertex of the second
graph. In this paper, we study the super edge-magic deficiencies of a wheel
minus an edge and join products of a path, a star, and a cycle, respectively,
with isolated vertices.Comment: 11 page
Langford sequences and a product of digraphs
Skolem and Langford sequences and their many generalizations have
applications in numerous areas. The -product is a generalization of
the direct product of digraphs. In this paper we use the -product
and super edge-magic digraphs to construct an exponential number of Langford
sequences with certain order and defect. We also apply this procedure to
extended Skolem sequences.Comment: 10 pages, 6 figures, to appear in European Journal of Combinatoric
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