Affiliation:
1. Department of Mathematics ETH Zürich Zürich Switzerland
2. School of Mathematical Sciences Tel Aviv University Tel Aviv Israel
3. Department of Mathematical Sciences, Mellon College of Science Carnegie Mellon University Pittsburgh Pennsylvania USA
Abstract
AbstractWe propose the following extension of Dirac's theorem: if is a graph with vertices and minimum degree , then in every orientation of there is a Hamilton cycle with at least edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree guarantees a Hamilton cycle with at least edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability is above the Hamiltonicity threshold, then, with high probability, in every orientation of there is a Hamilton cycle with edges oriented in the same direction.
Subject
Geometry and Topology,Discrete Mathematics and Combinatorics