Abstract

This paper focuses on anti-magic labellings of planar graphs of type (a, b, c) introduced by Lih (Util. Math. 24 (1983), 165–197). We show that a certain class of planar graphs defined using the complete graphs and a class of planar graphs defined using the complete bipartite graphs are (1, 1, 0) and (1, 1, 1)-anti-magic under certain mild conditions.

Abstract

This paper considers the problem of unicasting in wireless ad hoc networks. Unicasting is the problem of finding a route between a source and a destination and forwarding the message from the source to the destination. In theory, models that have been used oversimplify the problem of route discovery in ad hoc networks. The achievement of this paper is threefold. First we use a more general model in which nodes can have different transmission and interference ranges and we present a new routing algorithm for wireless ad hoc networks that has several nice features. We then combine our algorithm with that of known greedy algorithms to arrive at an average case efficient routing algorithm in the situation that GPS information is available. Finally we show how to schedule unicast traffic between a set of source-destination pairs by providing a proper vertex coloring of the nodes in the wireless ad hoc network. Our coloring algorithm achieves a O(Δ)–coloring that is locally distinct within the 2-hop neighborhood of any node.