An Analysis of an Optimal Bit Complexity Randomised Distributed Vertex Colouring Algorithm ⋆ (Extended Abstract) Y. M´ etivier, J.M. Robson, N. Saheb-Djahromi and A. Zemmari Universit´ e de Bordeaux - LaBRI 351 cours de la Lib´ eration, 33405 Talence, France {metivier, robson, saheb, zemmari}@labri.fr 1 Introduction 1.1 The Problem Let G =(V,E) be a simple undirected graph. A vertex colouring of G assigns colours to each vertex in such a way that neighbours have different colours. In this paper we discuss how efficient (time and bits) vertex colouring may be accomplished by exchange of bits between neighbouring vertices. The dis- tributed complexity of vertex colouring is of fundamental interest for the study and analysis of distributed computing. Usually, the topology of a distributed sys- tem is modelled by a graph and paradigms of distributed systems are encoded by classical problems in graph theory; among these classical problems one may cite the problems of vertex colouring, computing a maximal independent set, finding a vertex cover or finding a maximal matching. Each solution to one of these problems is a building block for many distributed algorithms: symmetry breaking, topology control, routing, resource allocation. 1.2 The Model The Network. We consider the standard message passing model for distributed computing. The communication model consists of a point-to-point communica- tion network described by a simple undirected graph G =(V,E) where the vertices of V represent network processors and the edges represent bidirectional communication channels. Processes communicate by message passing: a process sends a message to another by depositing the message in the corresponding chan- nel. We assume the system synchronous and synchronous wake up of processors: processors have access to a global clock and all processors start the algorithm at the same time. Time Complexity. A round (cycle) of each processor is composed of the fol- lowing three steps: 1. Send messages to (some of) the neighbours, 2. Receive messages from (some of) the neighbours, 3. Perform some local computation. As ⋆ This work was supported by grant No ANR-06-SETI-015-03 awarded by Agence Nationale de la Recherche