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...

where E(v) is the set of edges that have v as an end-point. The total labelling λ of G is vertex-magic if every vertex has the same weight, and the graph G is vertexmagic if a vertex-magic total labelling of G exists. Magic labellings of graphs were introduced by Sedlácěk [5] in 1963, and vertex-magic total labellings first appeared in 2002 in [4]. For a dynamic survey of various forms of graph...

We introduce two notions, (i)oriented labelling and (ii) oriented labelling, to explore frequency assignment problem under halfduplex setting and compute bounds of oriented labelling of special classes of graphs : trees, bipartite graphs and planar graphs. We do study an oriented version of clique cover problem as well. The frequency assignment problem (FAP) arises in wireless communication sys...

A total edge-irregular k-labelling ξ : V (G) ∪ E(G) → {1, 2, . . . , k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-lab...

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 ...

We investigate the existence of edge-magic labellings of countably infinite graphs by abelian groups. We show for that for a large class of abelian groups, including the integers Z, there is such a labelling whenever the graph has an infinite set of disjoint edges. A graph without an infinite set of disjoint edges must be some subgraph of H + I, where H is some finite graph and I is a countable...

We introduce the concept of constant 2-labelling of a vertex-weighted graph and show how it can be used to obtain perfect weighted coverings. Roughly speaking, a constant 2-labelling of a vertex-weighted graph is a black and white colouring of its vertex set which preserves the sum of the weights of black vertices under some automorphisms. We study constant 2-labellings on four types of vertex-...

We exhibit a graceful labelling for each generalised Petersen graph P8t,3 with t ≥ 1. As an easy consequence, we obtain that for any fixed t the corresponding graph is the unique starter graph for a cyclic edgedecomposition of the complete graph K2t+1. Due to its extreme versatility, the technique employed looks promising for finding new graceful labellings, not necessarily involving generalise...

Given an undirected graph G, an L(h, k)-labelling of G assigns colors to vertices from the integer set {0, . . . , λh,k}, such that any two vertices vi and vj receive colors c(vi) and c(vj) satisfying the following conditions: i) if vi and vj are adjacent then |c(vi)− c(vj)| ≥ h; ii) if vi and vj are at distance two then |c(vi)− c(vj)| ≥ k. The aim of the L(h, k)-labelling problem is to minimiz...

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels o...