  A hamiltonian coloring c of a graph G of order p is an assignment of colors
to the vertices of G such that $D(u,v)+|c(u)-c(v)|\geq p-1$ for every two
distinct vertices u and v of G, where D(u,v) denoted the detour distance
between u and v. The value hc(c) of a hamiltonian coloring c is the maximum
color assigned to a vertex of G. The hamiltonian chromatic number, denoted by
hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this
paper, we present a lower bound for the hamiltonian chromatic number of block
graphs and give a sufficient condition to achieve the lower bound. We
characterize symmetric block graphs achieving this lower bound. We present two
algorithms for optimal hamiltonian coloring of symmetric block graphs.
