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 G 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.