  In 2006 Bar{\'a}t and Thomassen conjectured that every planar
$4$-edge-connected $4$-regular simple graph of size divisible by three admits a
claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of
planar graphs with edge-connectivity $4$ which the smallest one contains $24$
vertices. In this note, we first give a smaller counterexample having only $18$
vertices and next construct a family of planar $4$-connected essentially
$6$-edge-connected $4$-regular simple graphs of size divisible by three with no
claw-decompositions. This result provides the sharpness for two known results
which say that every $5$-edge-connected graph of size divisible by three admits
a claw-decomposition if it is essentially $6$-edge-connected or planar.
