On vertices enforcing a Hamiltonian cycle

On vertices enforcing a Hamiltonian cycle

A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar gra...

Placing Specified Vertices at Precise Locations on a Hamiltonian Cycle

Sharp minimum degree and degree sum conditions are proven for the existence of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs.

Distributing vertices along a Hamiltonian cycle in Dirac graphs

Distributing vertices on Hamiltonian cycles

Let G be a graph of order n and 3 ≤ t ≤ n4 be an integer. Recently, Kaneko and Yoshimoto provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this paper, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order ...

Precise location of vertices on Hamiltonian cycles

Given k ≥ 2 fixed positive integers p1, p2, . . . , pk−1 ≥ 2, and k vertices {x1, x2, . . . , xk}, let G be a simple graph of sufficiently large order n. It is proved that if δ(G) ≥ (n+ 2k− 2)/2, then there is a Hamiltonian cycle C of G containing the vertices in order such that the distance along C is dC (xi, xi+1) = pi for 1 ≤ i ≤ k − 1. Also, let {(xi, yi)|1 ≤ i ≤ k} be a set of k disjoint p...