312 research outputs found
Vertex-magic Labeling of Trees and Forests
A vertex-magic total labeling of a graph G(V,E) is a one-to-one map λ from E ∪ V onto the integers {1, 2, . . . , |E| + |V|} such that
λ(x) + Σ λ(xy) where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings of these graphs. We pay special attention to the case of so-called galaxies, forests in which every component tree is a star
Group twin coloring of graphs
For a given graph , the least integer such that for every
Abelian group of order there exists a proper edge labeling
so that for each edge is called the \textit{group twin
chromatic index} of and denoted by . This graph invariant is
related to a few well-known problems in the field of neighbor distinguishing
graph colorings. We conjecture that for all graphs
without isolated edges, where is the maximum degree of , and
provide an infinite family of connected graph (trees) for which the equality
holds. We prove that this conjecture is valid for all trees, and then apply
this result as the base case for proving a general upper bound for all graphs
without isolated edges: , where
denotes the coloring number of . This improves the best known
upper bound known previously only for the case of cyclic groups
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
Enumerating super edge-magic labelings for the union of non-isomorphic graphs
A super edge-magic labeling of a graph G=(V,E) of order p and size q is a bijection f:V ∪E→{i}p+qi=1 such that: (1) f(u)+f(uv)+f(v)=k for all uv∈E; and (2) f(V )={i}pi=1. Furthermore, when G is a linear forest, the super edge-magic labeling of G is called strong if it has the extra property that if uv∈E(G) , u′,v′ ∈V (G) and dG (u,u′ )=dG (v,v′ )<+∞, then f(u)+f(v)=f(u′ )+f(v′ ). In this paper we introduce the concept of strong super edge-magic labeling of a graph G with respect to a linear forest F, and we study the super edge-magicness of an odd union of nonnecessarily isomorphic acyclic graphs. Furthermore, we find exponential lower bounds for the number of super edge-magic labelings of these unions. The case when G is not acyclic will be also considered.Preprin
On cordial labeling of hypertrees
Let be a vertex labeling of a hypergraph
. This labeling induces an~edge labeling of defined by
, where the sum is taken modulo . We say that is
-cordial if for all the number of vertices with
label differs by at most from the number of vertices with label and
the analogous condition holds also for labels of edges. If admits a
-cordial labeling then is called -cordial. The existence of
-cordial labelings has been investigated for graphs for decades.
Hovey~(1991) conjectured that every tree is -cordial for every .
Cichacz, G\"orlich and Tuza~(2013) were first to investigate the analogous
problem for hypertrees, that is, connected hypergraphs without cycles. The main
results of their work are that every -uniform hypertree is -cordial for
every and that every hypertree with or odd is -cordial.
Moreover, they conjectured that in fact all hypertrees are -cordial. In this
article, we confirm the conjecture of Cichacz et al. and make a step further by
proving that for every hypertree is -cordial.Comment: 12 page
Enumerating super edge-magic labelings for some types of path-like trees
The main goal of this paper is to use a variation of the Kronecker product
of matrices in order to obtain lower bounds for the number of non isomorphic
super edge-magic labelings of some types of path-like trees. As a corollary of the
results obtained here we also obtain lower bounds for the number of harmonious
labelings of the same type of trees.Postprint (published version
Antimagic Labeling of Forests
An antimagic labeling of a graph G(V,E) is a bijection f mapping from E to the set {1,2,…, |E|}, so that for any two different vertices u and v, the sum of f(e) over all edges e incident to u, and the sum of f(e) over all edges e incident to v, are distinct. We call G antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree.
It was proved by Kaplan, Lev, and Roditty in 2009, and by Liang, Wong, and Zhu in 2014 that every tree with at most one vertex of degree two is antimagic. A major tool used in the proof is the zero-sum partition introduced by Kaplan, Lev, and Roditty in 2009. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree two is also antimagic
- …