Richard Hammack

*More on direct product cancellation of graphs *

**Abstract**. The direct (or tensor) product of graphs obeys a limited cancellation property. Lovász proved that if *C *
has an odd cycle then * A* × *C* = *B* × *C* if and only if* A* = *B*, but
cancellation can fail if *C* is bipartite. We investigate the ways
cancellation can fail. Given a graph *A* and a bipartite graph *C*,
we classify the graphs *B* for which * A* × *C* = *B* × *C*.
Further, we give exact conditions on *A* that guarantee *A* × *C* = *B* × *C* implies * A* = *B*.
Combined with Lovász’s result, this characterizes the situations in which cancellation holds or fails.