Constructing labeling schemes through universal matrices

Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs labels the vertices of all graphs in such that for every graph and every two vertices u,v G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in . This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let be a family of connected graphs of size at most n and let denote the collection of graphs of size at most n, such that each graph in is composed of a disjoint union of some graphs in . We first investigate methods for translating f-labeling schemes for to f-labeling schemes for . In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family , one can construct an f-labeling scheme of size g(n)+loglogn+O(1) for . We also show that in several cases, the above mentioned extra additive term of loglogn+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size logn+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5logn+O(1) and a lower bound of 4/3logn-O(1) for the size of any such labeling scheme. Keywords: Labeling schemes, Universal graphs, Universal matrices.