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All-to-all communication occurs in many important applications in parallel processing. In this paper, we study the all-to-all broadcast number (the shortest time needed to complete the all-to-all broadcast) of Cartesian product of graphs under the assumption that: each vertex can use all of its links at the same time, and each communication link is half duplex and can carry only one message at a unit of time. We give upper and lower bounds for the all-to-all broadcast number of Cartesian product of graphs and give formulas for the all-to-all broadcast numbers of some classes of graphs, such as the Cartesian product of two cycles, the Cartesian product of a cycle with a complete graph of odd order, the Cartesian product of two complete graphs of odd order, and the hypercube Q 2 n under this model. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph and let be a positive integer. Consider the following two-person game which is played on : Alice and Bob alternate turns. A move consists of selecting an unlabeled vertex of and assigning it a number from satisfying the condition that, for all is labeled by the number previously, if , then , and if , then . Alice wins if all the vertices of are successfully labeled. Bob wins if an impasse is reached before all vertices in the graph are labeled. The game -labeling number of a graph is the least for which Alice has a winning strategy. We use to denote the game -labeling number of in the game Alice plays first, and use to denote the game -labeling number of in the game Bob plays first. In this paper, we study the game -labeling numbers of graphs. We give formulas for and , and give formulas for for those with . [Copyright &y& Elsevier]