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)
Nearby in the stack
+
∣c(u)−c(v)∣
≥
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 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.