Abstract:
The thesis deals with the problem of labeling the vertices, edges and faces of
a plane graphs by the consecutive integers in such a way that the label of a face
and the labels of the vertices and edges surrounding that face all together add up
to a weight of that face. If these face weights form an arithmetic progression with
common difference d then the labeling is called d-antimagic. Such a labeling is called
super if the smallest possible labels appear on the vertices.
The thesis examines the existence of such labelings for toroidal fullerenes, gener-
alized prism and disjoint union of generalized prisms.
The toroidal fullerene is a 2-colorable cubic graph, there exist a 1-factor (perfect
matching) and a 2-factor (a collection of n cycles on 2m vertices each). First we
label the vertices of toroidal fullerene and then we label the edges of a 1-factor by
consecutive integers and then in successive steps we label the edges of 2m-cycles
(respectively 2n-cycles) in a 2-factor by consecutive integers. This technique allows
us to construct super d-antimagic labelings of type (1, 1, 1) of toroidal fullerenes for
several values of d.
We consider the generalized prism as a collection of two classes of cycles: the
main cycles and the middle cycles. To label the main cycles and the middle cycles
we use the super (a, d)-edge-antimagic total and (a, d)-edge-antimagic total labelings
and combine these labelings to a resulting super d-antimagic labeling of type (1, 1, 1).
The disjoint union of generalized prism can be considered as a collection of disjoint
union of main cycles and disjoint union of middle cycles. To label the disjoint union of
main and middle cycles we again use edge-antimagic total labelings and super edge-
antimagic total labelings. Combining these labelings we obtain a resulting super
d-antimagic labeling of type (1, 1, 1) for a given diference d.
