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

The first problem we deal with in this dissertation concerns long induced cycles in the d-dimensional cube, i.e. the graph on d-tuples of binary digits with pairs of vertices differing in exactly one coordinate as edges. Such cycles were introduced by Kautz as a type of error-checking codes (called the snake-in-the-box codes) for a certain analogue-to-digital conversion system. We give a simple...

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...

We define a notion of combinatorial labellings, and show that ∆02 is the largest boldface pointclass in which every set admits a combinatorial labelling.

We introduce the concept of an edge-colouring total k-labelling. This is a labelling of the vertices and the edges of a graph G with labels 1, 2, . . . , k such that the weights of the edges define a proper edge colouring of G. Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define χt(G) to be the smallest integer k for which G has an edge-colouring ...