  A graph K is multiplicative if a homomorphism from any product G x H to K
implies a homomorphism from G or from H. Hedetniemi's conjecture states that
all cliques are multiplicative. In an attempt to explore the boundaries of
current methods, we investigate strongly multiplicative graphs, which we define
as K such that for any connected graphs G,H with odd cycles C,C', a
homomorphism from $(G \times C') \cup (C \times H) \subseteq G \times H$ to K
implies a homomorphism from G or H.
  Strong multiplicativity of K also implies the following property, which may
be of independent interest: if G is non-bipartite, H is a connected graph with
a vertex h, and there is a homomorphism $\phi \colon G \times H \to K$ such
that $\phi(-,h)$ is constant, then H admits a homomorphism to K.
  All graphs currently known to be multiplicative are strongly multiplicative.
We revisit the proofs in a different view based on covering graphs and replace
fragments with more combinatorial arguments. This allows us to find new
(strongly) multiplicative graphs: all graphs in which every edge is in at most
square, and the third power of any graph of girth >12. Though more graphs are
amenable to our methods, they still make no progress for the case of cliques.
Instead we hope to understand their limits, perhaps hinting at ways to further
extend them.
