  Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$
of a graph $G$ is an assignment of colors (non-negative integers) to the
vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every
two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour
distance between $u$ and $v$ in $G$ which is the length of the longest path
between $u$ and $v$. The value \emph{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 min\{$hc(c)$\} taken over all hamiltonian
coloring $c$ of $G$. In this paper, we give a necessary and sufficient
condition to achieve a lower bound for the hamiltonian chromatic number of
block graphs given in [Theorem 1,On Hamiltonian Colorings of Block graphs, In:
Kaykobad, M., Petrechi, R., (eds.) WALCOM: Algorithms and Computation, LNCS:
9627, 28-39, 2016]. We present an algorithm for optimal hamiltonian coloring of
a special class of block graphs, namely $SDB(p/2)$ block graphs. We
characterize level-wise regular block graphs and extended star of blocks
achieving this lower bound.
