Existence of Δ λ ‐cycles and Δ λ ‐paths

A Cycle C of a graph G is called a D λ ‐cycle if every component of G − V ( C ) has order less than λ A D λ ‐path is defined analogously. D λ ‐cycles and D λ ‐paths were introduced by Veldman. Here a cycle C of a graph G is called a Δ λ ‐cycle if all vertices of G are at distance less than λ from a vertex of C . A Δ λ ‐path is defined analogously. In particular, in a connected graph, a Δ λ ‐cycle is a Δ λ ‐Cycle and a Δ λ ‐Path is a Δ‐path. Furthermore, a Δ 1 ‐cycle is a Hamilton cycle and a Δ 1 path is a Hamilton path. Necessary conditions and sufficient conditions are derived for graphs to have a Δ λ ‐cycle or Δ λ ‐path. The results are analogues of theorems on D λ ‐cycles and D λ ‐paths. In particular, a result of Chvátal and Erdös on Hamilton cycles and Hamiiton paths is generalized. A recent conjecture of Bondy and Fan is settled.