For a connected graph
of order
, the detour
distance
between
two vertices
and
in
is the length of a
longest
path in
. A Hamiltonian
labeling of
is a
function
such that
for every two
distinct vertices
and
of
. The value
of a Hamiltonian
labeling
of
is the maximum label (functional value) assigned to a vertex of
by
; while the Hamiltonian
labeling number
of
is the minimum value of Hamiltonian labelings of
.
Hamiltonian labeling numbers of some well-known classes of graphs
are determined. Sharp upper and lower bounds are established for
the Hamiltonian labeling number of a connected graph. The corona
of a graph
is the graph
obtained from
by adding exactly one pendant edge at each vertex of
. For each
integer
, let
be the set of connected
graphs
for which there
exists a Hamiltonian graph
of order
such that
. It
is shown that
for each
and that both bounds are sharp.
