Locating any two vertices on Hamiltonian cycles

TL;DR

This paper proves Enomoto's conjecture for large graphs, showing that under certain degree conditions, any two vertices can be located at a specific distance on a Hamiltonian cycle.

Contribution

The paper provides a proof of Enomoto's conjecture for sufficiently large graphs using advanced combinatorial tools.

Findings

Confirmed Enomoto's conjecture for large graphs

Established existence of Hamiltonian cycles with prescribed vertex distances

Applied Regularity and Blow-up Lemmas in the proof

Abstract

In this paper we give a proof of Enomoto's conjecture for graphs of sufficiently large order. Enomoto's conjecture 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 $d_C(x,y)=\lfloor \frac{n}{2}\rfloor$. The main tools of our proof are Regularity Lemma of Szemer\'edi and Blow-up Lemma of Koml\'os et al.