3 research outputs found
Starter sequences: generalizations and applications
In this thesis we introduce new types of starter sequences, pseudo-starter sequences,
starter-labellings, and generalized (extended) starter sequences. We apply these new
sequences to graph labeling. All the necessary conditions for the existence of starter,
pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences
are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k-
windmills are investigated. We show that all paths, cycles, and k-windmills can be
starter-labelled or minimum starter-labelled
Graceful labellings of new families of windmill and snake graphs
A function Æ’ is a graceful labelling of a graph G = (V,E) with m edges if Æ’ is an
injection ƒ : V (G) → {0, 1, 2, . . . ,m} such that each edge uv ∈ E is assigned the
label |ƒ(u) − ƒ(v)| ∈ {1, 2, . . . ,m}, and no two edge labels are the same. If a graph
G has a graceful labelling, we say that G itself is graceful. A variant is a near graceful
labelling, which is similar, except the co-domain of f is {0, 1, 2, . . . ,m + 1} and the
set of edge labels are either {1, 2, . . . ,m − 1,m} or {1, 2, . . . ,m − 1,m + 1}.
In this thesis, we prove any Dutch windmill with three pendant triangles is (near)
graceful, which settles Rosa’s conjecture for a new family of triangular cacti. Further,
we introduce graceful and near graceful labellings of several families of windmills.
In particular, we use Skolem-type sequences to prove (near) graceful labellings exist
for windmills with C₃ and C₄ vanes, and infinite families of 3,5-windmills and 3,6-windmills. Furthermore, we offer a new solution showing that the graph obtained
from the union of t 5-cycles with one vertex in common (Ctâ‚…
) is graceful if and only if
t ≡ 0,3 (mod 4) and near graceful when t ≡ 1, 2 (mod 4).
Also, we present a new sufficiency condition to obtain a graceful labelling for every
kCâ‚„â‚™ snake and use this condition to label every such snake for n = 1, 2, . . . , 6. Then,
we extend this result to cyclic snakes where the cycles lengths vary. Also, we obtain
new results on the (near) graceful labelling of cyclic snakes based on cycles of lengths
n = 6, 10, 14, completely solving the case n = 6