An infinite graph G is called dominating if its vertices can be labelled with natural numbers in such a way that for every function f : ω→ ω there is a ray in G whose sequence of labels eventually exceeds f . Conversely, G is called bounded if for every labelling of its vertices with natural numbers there exists a function f : ω→ω which eventually exceeds the labelling along any ray in G. This ...

Suppose G is a finite graph with vertex-set V (G) and edge-set E(G). An edge-magic total labelling on G is a one-to-one map λ from V (G)∪E(G) onto the integers 1, 2, . . . , |V (G) ∪ E(G)| with the property that, given any edge (x, y), λ(x) + λ(x, y) + λ(y) = k for some constant k. Such a labelling is called strong if the smallest labels appear on the vertices. In this paper, we investigate the...

The problem of finding a graceful labelling of a graph, or proving that the graph is not graceful, has previously been modelled as a CSP. A new and much faster CSP model of the problem is presented, with several new results for graphs whose gracefulness was previously unknown. Several classes of graph that are conjectured to be graceful only for small instances are investigated: after a certain...

The radio channel assignment problem can be cast as a graph coloring problem. Vertices correspond to transmitter locations and their labels (colors) to radio channels. The assignment of frequencies to each transmitter (vertex) must avoid interference which depends on the seperation each pair of vertices has. Two levels of interference are assumed in the problem we are concerned. Based on this c...

For applied scientists and engineers, graph theory is a strong vital tool for evaluating inventing solutions variety of issues. Graph extremely important in complex systems, particularly computer science. Many scientific areas use theory, including biological sciences, engineering, coding, operational research. A strategy the orthogonal labelling bipartite <math xmlns="http://www.w3.org/1998/Ma...

Abstract Graphs in real-world applications are typically dynamic which undergo rapid changes their topological structure over time by either adding or deleting edges vertices. However, it is challenging to design algorithms capable of supporting updates efficiently on graphs. In this article, we devise a parallel fully labelling method reflect graphs when answering shortest-path distance querie...

Let σ, τ : N→N be two sequences of natural numbers. We say that σ dominates τ if σ(n) τ(n) for every n greater than some n0 ∈ N. Now let G be an infinite graph, and assume the vertices of G are labelled with natural numbers, say by f : V (G)→N. Then each ray (one-way infinite path) R = x0x1 . . . in G gives rise to a sequence τR := (f(x0), f(x1), . . .); we say that σ dominates R if σ dominates...

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 ...