Vertex Coloring with Communication and Local Memory Constraints in Synchronous Broadcast Networks

This paper considers the broadcast/receive communication model in which message collisions and message conflicts can occur because processes share frequency bands. (A collision occurs when, during the same round, messages are sent to the same process by too many neighbors. A conflict occurs when a process and one of its neighbors broadcast during the same round.) More precisely, the paper considers the case where, during a round, a process may either broadcast a message to its neighbors or receive a message from at most m of them. This captures communication-related constraints or a local memory constraint stating that, whatever the number of neighbors of a process, its local memory allows it to receive and store at most m messages during each round. The paper defines first the corresponding generic vertex multi-coloring problem (a vertex can have several colors). It focuses then on tree networks, for which it presents a lower bound on the number of colors K that are necessary (namely, \(K=\lceil \frac{\varDelta }{m}\rceil +1\), where \(\varDelta \) is the maximal degree of the communication graph), and an associated coloring algorithm, which is optimal with respect to K.