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.

[English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (694kB)

Abstract

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.

Actions (login required)

View Item