Coding the vertexes of a graph

Given a graph
G
of
n
nodes. We wish to assign to each node
i(i = 1, 2, \cdots n)
a unique binary code
c_{i}
of length
m
such that, if we denote the Hannuing distance between
c_{i}
and
c_{j}
as
H(c_{i}, c_{j})
, then
H(c_{i}, c_{j})\leq T
if nodes
i
and
j
are adjacent (i.e., connected by a single branch), and
H(c_{i}, c_{j}) \geq T+1
otherwise. If such a code exists, then we say that
G
is doable for the value of
T
and tn associated with this code. In this paper we prove various properties relevent to these codes. In particular we prove 1) that for every graph
G
there exists an
m
and
T
such that
G
is doable, 2) for every value of
T
there exists a graph
G̃
which is not
T
doable, 3) if
G
is
T'
doable, then it is
T'+ 2p
doable for
p = 0, 1, 2, \cdots
, and is doable for all
T \geq 2T'
if
T'
is odd, and is doable for all
T \geq 2T' + 1
if
T'
is even. In theory, the code can be synthesized by employing integer linear programming where either
T
and/or
m
can be minimized; however, this procedure is computationally infeasible for values of
n
and
m
in the range of about
10
or greater.