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Abstract We say that a graphHdominates another graphH^′if the number of homomorphisms fromH^′to any graphGis dominated, in an appropriate sense, by the number of homomorphisms fromHtoG. We study the family of dominating graphs, those graphs with the property that they dominate all of their subgraphs. It has long been known that even-length paths are dominating in this sense and a result of Hatami implies that all weakly norming graphs are dominating. In a previous paper, we showed that every finite reflection group gives rise to a family of weakly norming, and hence dominating, graphs. Here we revisit this connection to show that there is a much broader class of dominating graphs. 

Abstract Sidorenko’s conjecture states that, for all bipartite graphs $$H$$, quasirandom graphs contain asymptotically the minimum number of copies of $$H$$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $$r$$-partite $$r$$-uniform hypergraph $$H$$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $$\textrm{ex}(n,H)$$, the maximum number of edges in an $$n$$-vertex $$H$$-free $$r$$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $$3$$-partite $$3$$-uniform tight cycles. 

Abstract A graph $$H$$ is common if the number of monochromatic copies of $$H$$ in a 2-edge-colouring of the complete graph $$K_n$$ is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree $$T$$ , there exists a triangle tree such that the graph obtained by adding $$T$$ as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most $$5$$ vertices yields a common graph.