2016/01/31 by Marcin Wrochna · 2 citations
Mathematics · #math.CO #msc:05C15 #msc:05C60 #msc:55U10
paper · pdf · doi:10.1016/j.jctb.2016.07.007
22 pages, 5 figures. Only minor changes. Accepted for publication in JCTb
arxiv created 2016/08/23 · arxiv updated 2016/08/24
A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques Kn are multiplicative. However, the only non-trivial graphs known to be multiplicative are K3, odd cycles, and still more generally, circular cliques Kp/q with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In particular, this gives the first multiplicative graphs of chromatic number higher than 4. Generalizing, in terms of the box complex, the topological insight behind existing proofs for odd cycles, we also give a different proof for circular cliques.