A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Štovíček and Thomason [20]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.

中文翻译：

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非二部 K-公共图

如果通过随机着色渐近最小化K _n的k边着色中H的单色副本的数量，则图H是k共同的。对于每个k，我们构造一个连通的非二分k公图。这解决了 Jagger、Štovíček 和 Thomason [20] 提出的问题。我们还证明了一个图H对每个k是k -common当且仅当H是 Sidorenko 并且H是对每个k是局部k -common当且仅当H是局部 Sidorenko。