In this paper, we show that the class of Extended Duplicate Graph of a Twig is Graceful and Skolem-graceful. AMS SUBJECT CLASSIFICATION: 05C78.

In 1991, Gnanajothi [4] proved that the path graph n P with n vertex and 1 n − edge is odd graceful, and the cycle graph m C with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph m n C P ∪ when 4, 6,8,10 m = and then we proved that the graph m n C P ∪ is odd graceful if m is even. Finall...

Let G(V,E) be a graph of order n and size m. A graceful labeling of G is an injection f : V (G) → {0, 1, 2, ...,m} such that, when each edge uv is assigned the label f(uv) = |f(u)− f(v)|, the resultant edge labels are distinct. We focus on general results in graceful labeling, and provide an affirmative answer to the following open problem: Can every connected graph be embedded as an induced su...

We present the first polynomial time construction procedure for generating graceful double-wheel graphs. A graph is graceful if its vertices can be labeled with distinct integer values from {0, ..., e}, where e is the number of edges, such that each edge has a unique value corresponding to the absolute difference of its endpoints. Graceful graphs have a range of practical application domains, i...

For $G$ a simple, connected graph, vertex labeling $f:V(G)\to \Z_+$ is called \emph{radio of $G$} if it satisfies $|f(u)-f(v)|\geq\diam(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The number the minimal span over radio labelings $G$. If bijective onto $\{1,2,\dots,|V(G)|\}$ exists, graceful} graph. We determine diameter 3 Hamming graphs and show that an infinite subset them graceful.

Programming languages are enormously diverse, both in their essential concepts and in their accidental aspects. This creates a problem when teaching programming. To let students experience the diversity of essential concepts, the students must also be exposed to an overwhelming variety of accidental and irrelevant detail: the accidental differences between the languages are likely to obscure th...

In this paper, we prove that the square graph of bistar Bn,n, the splitting graph of Bn,n and the splitting graph of star K1,n are graceful graphs. We also prove that the splitting graph and the shadow graph of bistar Bn,n admit odd graceful labeling.