Distributing vertices along a Hamiltonian cycle in Dirac graphs

Abstract

A graph G on n vertices is called a Dirac graph if it has a minimum degree of at least n / 2 . The distance dist G ( u , v ) is defined as the number of edges in a shortest path of G joining u and v . In this paper we show that in a Dirac graph G , for every small enough subset S of the vertices, we can distribute the vertices of S along a Hamiltonian cycle C of G in such a way that all but two pairs of subsequent vertices of S have prescribed distances (apart from a difference of at most 1) along C . More precisely we show the following. There are ω , n 0 > 0 such that if G is a Dirac graph on n ≥ n 0 vertices, d is an arbitrary integer with 3 ≤ d ≤ ω n / 2 and S is an arbitrary subset of the vertices of G with 2 ≤ | S | = k ≤ ω n / d , then for every sequence d i of integers with 3 ≤ d i ≤ d , 1 ≤ i ≤ k − 1 , there is a Hamiltonian cycle C of G and an ordering of the vertices of S , a 1 , a 2 , … , a k , such that the vertices of S are visited in this order on C and we have | dist C ( a i , a i + 1 ) − d i | ≤ 1 , for all but one 1 ≤ i ≤ k − 1 .