  A graph H is common if the number of monochromatic copies of H in a
2-edge-colouring of the complete graph is minimised by the random colouring.
Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every
graph is common. The conjectures by Erdos and by Burr and Rosta were disproved
by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new
examples for common graphs had not seen much progress since then, although very
recently, a few more graphs are verified to be common by the flag algebra
method or the recent progress on Sidorenko's conjecture.
  Our contribution here is to give a new class of tripartite common graphs. The
first example class is so-called triangle-trees, which generalises two theorems
by Sidorenko and answers a question by Jagger, \v{S}\v{t}ov\'i\v{c}ek, 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 five vertices give a
common graph.
