Adjacency labeling schemes and induced-universal graphs

We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n\mathrm{/}2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an $n\mathrm{/}2+O(\log n)$ bound of Moon. As a consequence, we obtain an induced-universal graph for $n$-vertex graphs containing only $O(2^{n\mathrm{/}2})$ vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs.