Adjacency Labeling Schemes and Induced-Universal Graphs

TY - JOUR

T1 - Adjacency Labeling Schemes and Induced-Universal Graphs

AU - Alstrup, Stephen

AU - Kaplan, Haim

AU - Thorup, Mikkel

AU - Zwick, Uri

PY - 2019

Y1 - 2019

N2 - We describe a way of assigning labels to the vertices of any undirected graph on up to n vertices, each composed of n/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/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/ 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.

AB - We describe a way of assigning labels to the vertices of any undirected graph on up to n vertices, each composed of n/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/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/ 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.

U2 - 10.1137/16M1105967

DO - 10.1137/16M1105967

M3 - Journal article

SN - 0895-4801

VL - 33

SP - 116

EP - 137

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -