45 research outputs found
Antimagic Labelings of Caterpillars
A -antimagic labeling of a graph is an injection from to
such that all vertex sums are pairwise distinct, where
the vertex sum at vertex is the sum of the labels assigned to edges
incident to . We call a graph -antimagic when it has a -antimagic
labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel
conjectured that every simple connected graph other than is antimagic,
but the conjecture is still open even for trees. Here we study -antimagic
labelings of caterpillars, which are defined as trees the removal of whose
leaves produces a path, called its spine. As a general result, we use
constructive techniques to prove that any caterpillar of order is -antimagic. Furthermore, if is a caterpillar with a
spine of order , we prove that when has at least leaves or consecutive vertices of degree at
most 2 at one end of a longest path, then is antimagic. As a consequence of
a result by Wong and Zhu, we also prove that if is a prime number, any
caterpillar with a spine of order , or is -antimagic.Comment: 13 pages, 4 figure
On super (a, 1)-edge-antimagic total labelings of regular graphs
A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). An (a,d)-edge-antimagic total labeling of a graph with p vertices and q edges is a one-to-one mapping that takes the vertices and edges onto the integers 1,2…,p+q, so that the sum of the labels on the edges and the labels of their end vertices forms an arithmetic progression starting at a and having difference d. Such a labeling is called super if the p smallest possible labels appear at the vertices.
In this paper we prove that every even regular graph and every odd regular graph with a 1-factor are super (a,1)-edge-antimagic total. We also introduce some constructions of non-regular super (a,1)-edge-antimagic total graphs
Super (a, d)-Edge Antimagic Total Labeling of Connected Ferris Wheel Graph
Let G be a simple graph of order p and size q. Graph G is called an (a,d)-edge-antimagic totalifthereexistabijectionf :V(G)∪E(G)→{1,2,...,p+q}suchthattheedge-weights,w(uv)= f(u)+f(v)+f(uv); u, v ∈ V (G), uv ∈ E(G), form an arithmetic sequence with first term a and common difference d. Such a graph G is called super if the smallest possible labels appear on the vertices. In this paper we study super (a, d)-edge antimagic total properties of connected of Ferris Wheel F Wm,n by using deductive axiomatic method. The results of this research are a lemma or theorem. The new theorems show that a connected ferris wheel graphs admit a super (a, d)-edge antimagic total labeling for d = 0, 1, 2. It can be concluded that the result of this research has covered all feasible d. Key Words : (a, d)-edge antimagic vertex labeling, super (a, d)-edge antimagic total labeling, Ferris Wheel graph FWm,n.
Antimagic Labeling for Unions of Graphs with Many Three-Paths
Let be a graph with edges and let be a bijection from to
. For any vertex , denote by the sum of
over all edges incident to . If holds
for any two distinct vertices and , then is called an {\it antimagic
labeling} of . We call {\it antimagic} if such a labeling exists.
Hartsfield and Ringel in 1991 conjectured that all connected graphs except
are antimagic. Denote the disjoint union of graphs and by , and the disjoint union of copies of by . For an antimagic graph
(connected or disconnected), we define the parameter to be the
maximum integer such that is antimagic for all .
Chang, Chen, Li, and Pan showed that for all antimagic graphs , is
finite [Graphs and Combinatorics 37 (2021), 1065--1182]. Further, Shang, Lin,
Liaw [Util. Math. 97 (2015), 373--385] and Li [Master Thesis, National Chung
Hsing University, Taiwan, 2019] found the exact value of for special
families of graphs: star forests and balanced double stars respectively. They
did this by finding explicit antimagic labelings of and proving a
tight upper bound on for these special families. In the present
paper, we generalize their results by proving an upper bound on for
all graphs. For star forests and balanced double stars, this general bound is
equivalent to the bounds given in \cite{star forest} and \cite{double star} and
tight. In addition, we prove that the general bound is also tight for every
other graph we have studied, including an infinite family of jellyfish graphs,
cycles where , and the double triangle
Super (a,d)-edge-antimagic total labeling of connected Disc Brake graph
Super edge-antimagic total labeling of a graph with order and size , is a vertex labeling and an edge labeling such that the edge-weights, form an arithmetic sequence and for a>0 and , where is a label of vertex , is a label of vertex and is a label of edge . In this paper we discuss about super edge-antimagic total labelings properties of connective Disc Brake graph, denoted by . The result shows that a connected Disc Brake graph admit a super -edge antimagic total labeling for , , n is odd and . It can be concluded that the result has covered all the feasible
An Inductive Approach to Strongly Antimagic Labelings of Graphs
An antimagic labeling for a graph with edges is a bijection so that holds for any pair
of distinct vertices , where .
A strongly antimagic labeling is an antimagic labeling with an additional
condition: For any , if , then . A graph is strongly antimagic if it admits a strongly antimagic
labeling. We present inductive properties of strongly antimagic labelings of
graphs. This approach leads to simplified proofs that spiders and double
spiders are strongly antimagic, previously shown by Shang [Spiders are
antimagic, Ars Combinatoria, 118 (2015), 367--372] and Huang [Antimagic
labeling on spiders, Master's Thesis, Department of Mathematics, National
Taiwan University, 2015], and by Chang, Chin, Li and Pan [The strongly
antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020),
43--68], respectively. We fix a subtle error in [The strongly antimagic
labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68].
Further, we prove certain level-wise regular trees, cycle spiders and cycle
double spiders are all strongly antimagic