On Hamiltonicity of regular graphs with bounded second neighborhoods

Abstract

Let G(k) denote the set of connected k-regular graphs G, k ≥ 2, where the number of vertices at distance 2 from any vertex in G does not exceed k. Asratian ( 2006) showed (using other terminology) that a graph G ∈ G(k) is Hamiltonian if for each vertex u of G the subgraph induced by the set of vertices at distance at most 2 from u is 2-connected. We prove here that in fact all graphs in the sets G(3), G(4) and G(5) are Hamiltonian. We also prove that the problem of determining whether there exists a Hamilton cycle in a graph from G( 6) is NP-complete. Nevertheless we show that every locally connected graph G ∈ G(k), k ≥ 6, is Hamiltonian and that for each non-Hamiltonian cycle C in G there exists a cycleFinally, we note that all our conditions for Hamiltonicity apply to infinitely many graphs with large diameters.