Shorter labeling schemes for planar graphs

TLDR: In this paper, it was shown that planar graphs with n vertices admit a labeling scheme with labels of bit length (2 + o(1)) log n. This bound was improved to n4/3+o(1) by the authors of this paper.

Abstract: An adjacency labeling scheme for a given class of graphs is an algorithm that for every graph G from the class, assigns bit strings (labels) to vertices of G so that for any two vertices u, v, whether u and v are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with n vertices admit a labeling scheme with labels of bit length (2 + o(1)) log n. In this work we improve this bound by designing a labeling scheme with labels of bit length [MATH HERE]. In graph-theoretical terms, this implies an explicit construction of a graph on n4/3+o(1) vertices that contains all planar graphs on n vertices as induced subgraphs, improving the previous best upper bound of n2+o(1). Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time.