The -labelling of a graph is an abstraction of assigning integer frequencies to radio transmitters such that the transmitters that are one unit of distance apart receive frequencies that differ by at least two, and transmitters that are two units of distance apart receive frequencies that differ by at least one. The span of an -labelling is the difference between the largest and the smallest assigned frequency. The -labelling number of a graph , denoted by , is the least integer such that has an -labelling of span . A cactus graph is a connected graph in which every block is either an edge or a cycle. The goal of the problem is to show that for a cactus graph , where is the degree of . An optimal algorithm is also presented here to label the vertices of cactus graph using -labelling technique in time, where is the total number of vertices of the cactus graph.

An <i>L</i>(0,1)-labelling of a graph <i>G</i> is an assignment of nonnegative integers to the vertices of <i>G</i> such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an <i>L</i>(0,1)-labelling is the maximum label number assigned to any vertex of <i>G</i>. The <i>L</i>(0,1)-labelling number of a graph <i>G</i>, denoted by λ_0.1(<i>G</i>) is the least integer <i>k</i> such that <i>G</i> has an <i>L</i>(0,1)-labelling of span <i>k</i>. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by <i>L</i>(0,1)-labelling and have shown that, △-1≤λ_0.1(<i>G</i>)≤△ for a cactus graph, where △ is the degree of the graph <i>G</i>