DeBiasio, Louis and Faizullah, Safi and Khan, Imdadullah - Ore-degree threshold for the square of a Hamiltonian cycle

dmtcs:2127 - Discrete Mathematics & Theoretical Computer Science, February 2, 2015, Vol. 17 no. 1 (in progress)
Ore-degree threshold for the square of a Hamiltonian cycle

Authors: DeBiasio, Louis and Faizullah, Safi and Khan, Imdadullah

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.


Source : oai:HAL:hal-01218404v1
Volume: Vol. 17 no. 1 (in progress)
Section: Graph Theory
Published on: February 2, 2015
Submitted on: March 4, 2014
Keywords: Ore-degree,Hamiltonian cycle,cycle powers,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


Share

Browsing statistics

This page has been seen 23 times.
This article's PDF has been downloaded 67 times.