Let G = (V,E) be a simple graph with the vertex set V and the edge set E. G is a sum graph if there exists a labelling f of the vertices of G into distinct positive integers such that uv ∈ E if and only if f(w) = f(u) + f(v) for some vertex w ∈ V . Such a labelling f is called a sum labelling of G. The sum number σ(G) of G is the smallest number of isolated vertices which result in a sum graph ...

For positive integers k and d ≥ 2, a k-S(d, 1)-labelling of a graph G is a function on the vertex set of G, f : V (G) → {0, 1, 2, · · · , k− 1}, such that |f(u)− f(v)|k ≥ { d if dG(u, v) = 1; 1 if dG(u, v) = 2, where |x|k = min{|x|, k − |x|} is the circular difference modulo k. In general, this kind of labelling is called the S(d, 1)-labelling. The σdnumber of G, σd(G), is the minimum k of a k-...

Given an integer r > 0, let G r = (V; E) denote a graph consisting of a simple nite undirected connected nontrivial graph G together with r isolated vertices K r. Let L : V ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of V , (u; v) 2 E if and only...

A total labelling of a graph over an abelian group is a bijection from the set of vertices and edges onto the set of group elements. A labelling can be used to define a weight for each edge and for each vertex of finite degree. A labelling is edge-magic if all the edges have the same weight and vertex-magic if all the vertices are finite degree and have the same weight. We exhibit magic labelli...

Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every tree with at most 35 vertices allows a graceful labelling, hence we verify that the graceful tree conjecture is correct for trees with at most 35 vertices.

A graceful labelling of an undirected graph G with n edges is a one-to-one function from the set of vertices of G to the set {0, 1, 2, . . . , n} such that the induced edge labels are all distinct. An induced edge label is the absolute value of the difference between the two end-vertex labels. The Graceful Tree Conjecture states that all trees have a graceful labelling. In this survey we presen...

We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to repr...

The frequency assignment problem has been started from the discovery that transmitters, received the same or closely related frequencies, had interferences with one another. Nearly three decade back this problem has been modelled as a graph labelling problem. This labelling has several variations depending upon the type of assignment of frequency to transmitters. Here we shall discuss about som...

The (d,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we consider the list version of (d,1)-total labelling of graphs. Let G be a graph embedded in a surface with Euler characteristic ε whose maximum degree ∆(G) is sufficiently large. We prove that the (d,1)-total choosability C d,1(G) of G is at most ∆(G) + 2d.