22 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
Caterpillars have antimagic orientations
An antimagic labeling of a directed graph D with m arcs is a bijection from the set of arcs of D to {1, . . . , m} such that all oriented vertex sums of vertices in D are pairwise distinct, where the oriented vertex sum of a vertex u is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving
u. Hefetz, Mütze, and Schwartz [3] conjectured that every connected graph admits an antimagic orientation, where an
antimagic orientation of a graph G is an orientation of G which has an antimagic labeling. We use a constructive technique to prove that caterpillars, a well-known subclass of trees, have antimagic orientations.Peer ReviewedPostprint (published version
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