  An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such
that the sizes of any two color classes differ by at most one. In the paper, we
pose a conjecture that offers a gap-one bound for the smallest number of colors
needed to equitably color every block graph. In other words, the difference
between the upper and the lower bounds of our conjecture is at most one. Thus,
in some sense, the situation is similar to that of chromatic index, where we
have the classical theorem of Vizing and the Goldberg conjecture for
multigraphs. The results obtained in the paper support our conjecture. More
precisely, we verify it in the class of well-covered block graphs, which are
block graphs in which each vertex belongs to a maximum independent set. We also
show that the conjecture is true for block graphs, which contain a vertex that
does not lie in an independent set of size larger than two. Finally, we verify
the conjecture for some symmetric-like block graphs. In order to derive our
results we obtain structural characterizations of block graphs from these
classes.
