Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of n -vertex graph G is assigned a (1 + o(1)) log 2 -bit label and the labels two vertices u v are sufficient to determine if uv edge . This optimal up lower order term first such asymptotically result. An alternative, but equivalent, interpretation this result that, every positive integer , U with 1+o(1) induced subgraph These results generalize number other classes, including bounded genus graphs, apex-minor-free bounded-degree from minor closed families, k -planar graphs.