On direct product cancellation of graphs

The direct product of graphs obeys a limited cancellation property. Lovász proved that if CC has an odd cycle then A×C≅B×CA×C≅B×C if and only if A≅BA≅B, but cancellation can fail if CC is bipartite. This note investigates the ways cancellation can fail. Given a graph AA and a bipartite graph CC, we classify the graphs BB for which A×C≅B×CA×C≅B×C. Further, we give exact conditions on AA that guarantee A×C≅B×CA×C≅B×C implies A≅BA≅B. Combined with Lovász’s result, this completely characterizes the situations in which cancellation holds or fails.