28 research outputs found
On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture
International audienceThis paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices
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
Antimagic Labelings of Weighted and Oriented Graphs
A graph is - if for any vertex weighting
and any list assignment with there exists an edge labeling
such that for all , labels of edges are pairwise
distinct, and the sum of the labels on edges incident to a vertex plus the
weight of that vertex is distinct from the sum at every other vertex. In this
paper we prove that every graph on vertices having no or
component is -weighted-list-antimagic.
An oriented graph is - if there exists an
injective edge labeling from into such that the
sum of the labels on edges incident to and oriented toward a vertex minus the
sum of the labels on edges incident to and oriented away from that vertex is
distinct from the difference of sums at every other vertex. We prove that every
graph on vertices with no component admits an orientation that is
-oriented-antimagic.Comment: 10 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
Combinatorial nullstellensatz and its applications
In 1999, Noga Alon proved a theorem, which he called the Combinatorial Nullstellensatz, that gives an upper bound to the number of zeros of a multivariate polynomial. The theorem has since seen heavy use in combinatorics, and more specifically in graph theory. In this thesis we will give an overview of the theorem, and of how it has since been applied by various researchers. Finally, we will provide an attempt at a proof utilizing a generalized version of the Combinatorial Nullstellensatz of the GM-MDS Conjecture
Local Antimagic Coloring of Some Graphs
Given a graph , a bijection
is called a local antimagic labeling of if the vertex weight is distinct for all adjacent vertices. The vertex
weights under the local antimagic labeling of induce a proper vertex
coloring of a graph . The \textit{local antimagic chromatic number} of
denoted by is the minimum number of weights taken over all such
local antimagic labelings of . In this paper, we investigate the local
antimagic chromatic numbers of the union of some families of graphs, corona
product of graphs, and necklace graph and we construct infinitely many graphs
satisfying