  Graph burning is a natural discrete graph algorithm inspired by the spread of
social contagion. Despite its simplicity, some open problems remain steadfastly
unsolved, notably the burning number conjecture, which says that every
connected graph of order $m^2$ has burning number at most $m$. Earlier, we
showed that the conjecture also holds for a path forest, which is disconnected,
provided each of its paths is sufficiently long. However, finding the least
sufficient length for this to hold turns out to be nontrivial. In this note, we
present our initial findings and conjectures that associate the problem to some
naturally impossibly burnable path forests. It is noteworthy that our problem
can be reformulated as a topic concerning sumset partition of integers.
