  In this paper, we consider some general properties of block graphs as well as
the equitable coloring problem in this class of graphs. In the first part we
establish the relation between two structural parameters for general block
graphs. We also give complete characterization of block graphs with given value
of parameter $\alpha_{\min}$. In the next part of the paper we confirm the
hypothesis for some subclass of GLS block graphs in which the problem of
EQUITABLE COLORING is unlikely to be polynomial time solvable.
  We give also an equitable $(n+2)$-algorithm for all GLS block graphs. As a by
product we prove that the equitable chromatic spectrum for the subclass of GLS
block graphs is gap-free.
