Nash-Williams and Chvátal conditions (1969 and 1972) are well known and classical sufficient conditions for a graph to contain a Hamiltonian cycle. In this paper, we add constraints, called
. A conflict is a pair of edges of the graph that cannot be both in a same Hamiltonian path or cycle. Given a graph
and a set of conflicts, we try to determine whether
contains such a Hamiltonian path or cycle without conflict. We focus in this paper on graphs in which each vertex is part of at most one conflict, called
. We propose Nash-Williams-type and Chvátal-type results in this context.