无平方数图是乘法图
Square-free graphs are multiplicative
摘要：
一个图K是无平方的，如果它不包含四圈作为子图。一个图K是乘法的，如果对所有的图G，H，GxH -> K蕴涵G -> K或H -> K.这里GxH是张量（或范畴）图积，G -> K表示存在从G到K的图同态。Hedetniemi猜想指出所有团K_n都是乘法的。然而，已知是乘法的唯一非平凡图是K_3，奇圈，更一般地，是2 <= p/q < 4的循环团$K_{p/q}$。我们没有取得进展的团，但表明，所有的平方免费图是乘法。特别地，这给出了第一个色数大于4的乘法图。推广，在盒子复杂的，拓扑洞察力背后现有的证明奇循环，我们也给出了一个不同的证明循环集团。
英文摘要：
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 K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ 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.
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