TY - JOUR

T1 - Robust Hamiltonicity of Dirac graphs

AU - Krivelevich, Michael

AU - Lee, Choongbum

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2014

Y1 - 2014

N2 - A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac’s theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability p, and prove that there exists a constant C such that if p ≥ C log n/n, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a (1 : b) Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias b is at most cn/log n for some absolute constant c > 0. Both of these results are tight up to a constant factor, and are proved under one general framework.

AB - A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac’s theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability p, and prove that there exists a constant C such that if p ≥ C log n/n, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a (1 : b) Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias b is at most cn/log n for some absolute constant c > 0. Both of these results are tight up to a constant factor, and are proved under one general framework.

UR - http://www.scopus.com/inward/record.url?scp=84924794798&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2014-05963-1

DO - 10.1090/S0002-9947-2014-05963-1

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AN - SCOPUS:84924794798

SN - 0002-9947

VL - 366

SP - 3095

EP - 3130

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -