  Given a graph $G$ and a partition $P$ of its vertex set, an independent
transversal (IT) is an independent set of $G$ that contains one vertex from
each block in $P$. Various sufficient conditions for the existence of an IT
have been established, and a common theme for many of them is that the block
sizes are sufficiently large compared to the maximum degree of $G$.
Consequently, there has been interest in constructing graphs with no IT which
demonstrate that these bounds on the block sizes are best possible. We describe
a simple systematic method for constructing vertex-partitioned graphs with
large block sizes and no IT. Unifying previous constructions, we use our method
to derive classical extremal constructions due to Jin (1992), Yuster (1997),
and Szab\'o and Tardos (2006) in streamlined fashion. For our new results, we
describe extremal constructions of minimal graphs with maximum degree two and
no IT, generalizing a result of Aharoni, Holzman, Howard, and Spr\"ussel
(2015). We construct a family of locally sparse graphs with no IT,
complementing an asymptotic result of Loh and Sudakov (2007). We describe new
and smaller counterexamples to a list coloring conjecture of Reed (1999), which
was originally disproved by Bohman and Holzman (2002). We disprove a conjecture
of Aharoni, Alon, and Berger (2016) about IT's in graphs without large induced
stars. We answer negatively a question of Aharoni, Holzman, Howard, and
Spr\"ussel (2015) about extremal graphs with no IT, but we also prove that a
useful variation of their question does hold.
