A proof of a conjecture about D_λ-paths in graphs with large neighborhood unions

Abstract

A path P of a graph G is called a D_λ-path if every component of G/V(P) has order less than λ. In particular, a D₁-path is a hamiltonian path. Given a graph G, let H₁H₂ be subgraphs of G, we define the distance between H₁ and H₂ in G, denoted by d(H₁,H₂), to be the length of a shortest path in G starting at a vertex of H₁ and ending at a vertex of H₂. If G contains two connected subgraphs of order λ at distance two, then NC2_λ(G) denotes min {|N(H₁)UN(H₂)| H₁ and H₂ are connected subgraphs of order λ with d(H₁,H₂) = 2}; otherwise, NC2;(G) = n - 2λ + 1. Bauer, Fan and Veldman proved that a graph G of order n contains a D_λ-path if G is 2-connected and NC2_{(G)⩾ (2n + 2)/3 - 2λ}. They also conjectured that G contains a D_λ-path if G is 2-connected and NC2_{i(G) ⩾(n + 3)/2 - 2λ}. In this paper, we prove this conjecture is true and the bound (n + 3)/2 - 2λ is best possible. In particular, for λ = 1, it also solves Lindquester's conjecture about hamiltonian paths (D₁-paths) in graphs with large neighborhood unions.