Graceful labellings of new families of windmill and snake graphs

Alkasasbeh, Ahmad (2021) Graceful labellings of new families of windmill and snake graphs. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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A function ƒ is a graceful labelling of a graph G = (V,E) with m edges if ƒ 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 (Ct₅ ) 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 kC₄ₙ 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.

Item Type: Thesis (Doctoral (PhD))
Item ID: 15271
Additional Information: Includes bibliographical references.
Keywords: graceful labellings, Skolem-type sequences
Department(s): Science, Faculty of > Mathematics and Statistics
Date: August 2021
Date Type: Submission
Digital Object Identifier (DOI):
Library of Congress Subject Heading: Graph theory; Algorithms; Windmills—Netherlands; Skolem, Th. (Thoralf), 1887-1963.

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