arXiv subjects

Ahmad H. Alkasasbeh

Publications and source records attributed to Ahmad H. Alkasasbeh.

3 recordsLinked to original sources

arXiv2021

Graceful labellings of variable windmills using Skolem sequences

In this paper, 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 $C_3$ and $C_4$ 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 ($C_5^t$) is graceful if and only if $t \equiv 0,3\!\!\pmod{4}$ and is near graceful when $t\equiv 1,2 \pmod{4}$.

math.CO↗

arXiv2021

Applying Skolem Sequences to Gracefully Label New Families of Triangular Windmills

A function $f$ is a \textit{graceful labelling} of a graph $G=(V,E)$ with $m$ edges if $f$ is an injection $f:V\mapsto \{0,1,2,\dots,m\}$ such that each edge $uv \in E$ is assigned the label $|f(u)-f(v)|$, and no two edge labels are the same. If a graph G has a graceful labelling, we say that $G$ itself is graceful. In this paper, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa's conjecture for a new family of triangular cacti.

math.CO↗

arXiv2021

Graceful Labellings of Various Cyclic Snakes

In this paper, we present a new sufficiency condition to obtain a graceful labelling for every $kC_{4n}$ snake and use this condition to label every such snake for $n=1,2,\ldots,6$. 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 $n=6$.

math.CO↗