AMO - Advanced Modeling and Optimization, Volume 11, Number 4, 2009 Edge Colouring of Cactus Graphs Nasreen Khan † , Anita Pal ‡ and Madhumangal Pal † † Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721 102, INDIA e-mail: mmpalvu@gmail.com ‡ Department of Mathematics, National Institute of Technology Durgapur, Durgapur-713209, West Bengal, INDIA. e-mail: anita.buie@gmail.com Abstract. Edge colouring of an undirected graph G =(V,E) is assigning a colour to each edge e ∈ E so that any two edges having end-vertex in common have different colours. That is, the edge colouring problem asks for assigning colours from a minimum number of colours to edges of a graph such that no two edges with the same colour are incident to the same node. The minimum number of colours required for an edge colouring of G is denoted by χ ′ (G). A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we colour the edges of a cactus graph with minimum number of colours. Keywords: Graph colouring; edge colouring; cactus graph. AMS Subject Classifications: 68Q22, 68Q25, 68R10. 1 Introduction Cactus graph is a connected graph in which every block is either a cycle or an edge, in other words, no edge belongs to more than one cycle. Cactus graph have extensively studied and used as models for many real world problems. This graph is one of the most useful discrete mathematical structure for modelling problem arising in the real world. It has many applications 1 AMO - Advanced Modeling and Optimization, ISSN: 1841-4311. 407