On tripartite common graphs

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