Inspired by graceful labelings and total labelings of graphs, we introduce the idea of total difference
labelings. A
-total
labeling of a graph
is an assignment of
distinct labels to the edges and vertices of a graph so that adjacent vertices,
incident edges, and an edge and its incident vertices receive different labels. A
-total difference
labeling of a graph
is a
function
from the set
of edges and vertices of
to the set
that is
a
-total labeling
of
and for which
for any two
adjacent vertices
and
of
with incident
edge
. The least
positive integer
for which
has a
-total
difference labeling is its total difference chromatic number,
. We
determine the total difference chromatic number of paths, cycles, stars, wheels, gears
and helms. We also provide bounds for total difference chromatic numbers of
caterpillars, lobsters, and general trees.
