# Locating pairs of vertices on Hamiltonian cycles

Time: 14:30 -- Location: LRI, 435, salle des theses

We first introduce some of our recent results that generalises Dirac's theorem in Hamiltonian graph theory. Then we will focus on the following conjecture of Enomoto that states that, if \(G\) is a graph of order \(n\) with minimum degree \(delta(G)geq frac{n}{2}+1\), then for any pair of vertices \(x\), \(y\) in \(G\), there is a Hamiltonian cycle \(C\) of \(G\) such that \(d_C(x,y)=lfloor frac{n}{2}rfloor\). Weihua He, Qiang He and I gave a proof of Enomoto's conjecture for graphs of sufficiently large order. The main tools of our proof are Regularity Lemma of Szemer'{e}di and Blow-up Lemma of Koml'{o}s et al..